| L(s) = 1 | − 2·4-s + 12·7-s + 20·13-s − 16-s + 12·19-s − 14·25-s − 24·28-s − 16·31-s + 4·37-s + 16·43-s + 78·49-s − 40·52-s + 44·61-s + 8·64-s + 52·73-s − 24·76-s − 8·79-s + 240·91-s − 32·97-s + 28·100-s − 28·103-s + 40·109-s − 12·112-s + 32·124-s + 127-s + 131-s + 144·133-s + ⋯ |
| L(s) = 1 | − 4-s + 4.53·7-s + 5.54·13-s − 1/4·16-s + 2.75·19-s − 2.79·25-s − 4.53·28-s − 2.87·31-s + 0.657·37-s + 2.43·43-s + 78/7·49-s − 5.54·52-s + 5.63·61-s + 64-s + 6.08·73-s − 2.75·76-s − 0.900·79-s + 25.1·91-s − 3.24·97-s + 14/5·100-s − 2.75·103-s + 3.83·109-s − 1.13·112-s + 2.87·124-s + 0.0887·127-s + 0.0873·131-s + 12.4·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{12} \cdot 11^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{12} \cdot 11^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(271.6505895\) |
| \(L(\frac12)\) |
\(\approx\) |
\(271.6505895\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( ( 1 - T )^{12} \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} + 5 T^{4} + p^{2} T^{6} + p^{3} T^{8} - 3 p^{3} T^{10} - 3 p^{2} T^{12} - 3 p^{5} T^{14} + p^{7} T^{16} + p^{8} T^{18} + 5 p^{8} T^{20} + p^{11} T^{22} + p^{12} T^{24} \) |
| 5 | \( 1 + 14 T^{2} + 101 T^{4} + 98 p T^{6} + 2846 T^{8} + 20574 T^{10} + 123321 T^{12} + 20574 p^{2} T^{14} + 2846 p^{4} T^{16} + 98 p^{7} T^{18} + 101 p^{8} T^{20} + 14 p^{10} T^{22} + p^{12} T^{24} \) |
| 13 | \( ( 1 - 10 T + 83 T^{2} - 500 T^{3} + 2621 T^{4} - 870 p T^{5} + 44436 T^{6} - 870 p^{2} T^{7} + 2621 p^{2} T^{8} - 500 p^{3} T^{9} + 83 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 17 | \( 1 + 90 T^{2} + 4261 T^{4} + 146590 T^{6} + 4020830 T^{8} + 89661290 T^{10} + 1660695689 T^{12} + 89661290 p^{2} T^{14} + 4020830 p^{4} T^{16} + 146590 p^{6} T^{18} + 4261 p^{8} T^{20} + 90 p^{10} T^{22} + p^{12} T^{24} \) |
| 19 | \( ( 1 - 6 T + 106 T^{2} - 530 T^{3} + 4853 T^{4} - 19396 T^{5} + 121592 T^{6} - 19396 p T^{7} + 4853 p^{2} T^{8} - 530 p^{3} T^{9} + 106 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 23 | \( 1 + 168 T^{2} + 13726 T^{4} + 723160 T^{6} + 27775475 T^{8} + 839711408 T^{10} + 21022991420 T^{12} + 839711408 p^{2} T^{14} + 27775475 p^{4} T^{16} + 723160 p^{6} T^{18} + 13726 p^{8} T^{20} + 168 p^{10} T^{22} + p^{12} T^{24} \) |
| 29 | \( 1 + 158 T^{2} + 12523 T^{4} + 676610 T^{6} + 28938587 T^{8} + 1051503368 T^{10} + 32911943606 T^{12} + 1051503368 p^{2} T^{14} + 28938587 p^{4} T^{16} + 676610 p^{6} T^{18} + 12523 p^{8} T^{20} + 158 p^{10} T^{22} + p^{12} T^{24} \) |
| 31 | \( ( 1 + 8 T + 110 T^{2} + 688 T^{3} + 4823 T^{4} + 26760 T^{5} + 147540 T^{6} + 26760 p T^{7} + 4823 p^{2} T^{8} + 688 p^{3} T^{9} + 110 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 37 | \( ( 1 - 2 T + 91 T^{2} - 314 T^{3} + 5891 T^{4} - 17852 T^{5} + 259334 T^{6} - 17852 p T^{7} + 5891 p^{2} T^{8} - 314 p^{3} T^{9} + 91 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 41 | \( 1 + 178 T^{2} + 13639 T^{4} + 560026 T^{6} + 12799187 T^{8} + 170859508 T^{10} + 2999880794 T^{12} + 170859508 p^{2} T^{14} + 12799187 p^{4} T^{16} + 560026 p^{6} T^{18} + 13639 p^{8} T^{20} + 178 p^{10} T^{22} + p^{12} T^{24} \) |
| 43 | \( ( 1 - 8 T + 152 T^{2} - 1072 T^{3} + 11384 T^{4} - 75288 T^{5} + 582042 T^{6} - 75288 p T^{7} + 11384 p^{2} T^{8} - 1072 p^{3} T^{9} + 152 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 47 | \( 1 + 332 T^{2} + 57032 T^{4} + 6587500 T^{6} + 563615924 T^{8} + 37403435100 T^{10} + 1967941380078 T^{12} + 37403435100 p^{2} T^{14} + 563615924 p^{4} T^{16} + 6587500 p^{6} T^{18} + 57032 p^{8} T^{20} + 332 p^{10} T^{22} + p^{12} T^{24} \) |
| 53 | \( 1 + 382 T^{2} + 66487 T^{4} + 7011646 T^{6} + 511558763 T^{8} + 29260051156 T^{10} + 1541317998842 T^{12} + 29260051156 p^{2} T^{14} + 511558763 p^{4} T^{16} + 7011646 p^{6} T^{18} + 66487 p^{8} T^{20} + 382 p^{10} T^{22} + p^{12} T^{24} \) |
| 59 | \( 1 + 388 T^{2} + 79864 T^{4} + 11127724 T^{6} + 1159312964 T^{8} + 94699812532 T^{10} + 6212827842398 T^{12} + 94699812532 p^{2} T^{14} + 1159312964 p^{4} T^{16} + 11127724 p^{6} T^{18} + 79864 p^{8} T^{20} + 388 p^{10} T^{22} + p^{12} T^{24} \) |
| 61 | \( ( 1 - 22 T + 418 T^{2} - 5566 T^{3} + 62885 T^{4} - 613252 T^{5} + 5052104 T^{6} - 613252 p T^{7} + 62885 p^{2} T^{8} - 5566 p^{3} T^{9} + 418 p^{4} T^{10} - 22 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 67 | \( ( 1 + 196 T^{2} + 64 T^{3} + 21428 T^{4} + 1352 T^{5} + 1671734 T^{6} + 1352 p T^{7} + 21428 p^{2} T^{8} + 64 p^{3} T^{9} + 196 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 71 | \( 1 + 72 T^{2} + 12670 T^{4} + 721048 T^{6} + 105137939 T^{8} + 4570036592 T^{10} + 536793730748 T^{12} + 4570036592 p^{2} T^{14} + 105137939 p^{4} T^{16} + 721048 p^{6} T^{18} + 12670 p^{8} T^{20} + 72 p^{10} T^{22} + p^{12} T^{24} \) |
| 73 | \( ( 1 - 26 T + 622 T^{2} - 9206 T^{3} + 127301 T^{4} - 1316864 T^{5} + 12781136 T^{6} - 1316864 p T^{7} + 127301 p^{2} T^{8} - 9206 p^{3} T^{9} + 622 p^{4} T^{10} - 26 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 79 | \( ( 1 + 4 T + 106 T^{2} - 596 T^{3} + 1187 T^{4} - 48536 T^{5} + 293876 T^{6} - 48536 p T^{7} + 1187 p^{2} T^{8} - 596 p^{3} T^{9} + 106 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 83 | \( 1 + 260 T^{2} + 54704 T^{4} + 7982020 T^{6} + 1013286620 T^{8} + 104330092020 T^{10} + 9478839943518 T^{12} + 104330092020 p^{2} T^{14} + 1013286620 p^{4} T^{16} + 7982020 p^{6} T^{18} + 54704 p^{8} T^{20} + 260 p^{10} T^{22} + p^{12} T^{24} \) |
| 89 | \( 1 + 422 T^{2} + 99085 T^{4} + 17225714 T^{6} + 2365521806 T^{8} + 269042562326 T^{10} + 25988928108833 T^{12} + 269042562326 p^{2} T^{14} + 2365521806 p^{4} T^{16} + 17225714 p^{6} T^{18} + 99085 p^{8} T^{20} + 422 p^{10} T^{22} + p^{12} T^{24} \) |
| 97 | \( ( 1 + 16 T + 459 T^{2} + 3698 T^{3} + 62861 T^{4} + 231678 T^{5} + 5290852 T^{6} + 231678 p T^{7} + 62861 p^{2} T^{8} + 3698 p^{3} T^{9} + 459 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.26241433539667885574442666574, −2.16922431261332485128108121999, −2.05866325366553463023917601241, −2.04702207310873682330929801011, −1.97955967923853686672077331736, −1.94803418864404580734222265455, −1.79682071162444831017703393448, −1.57029505551627597010535868641, −1.54883400628448331681135671755, −1.53290341171665545339208318797, −1.47198583317822233863059576074, −1.44328432241407339075380971020, −1.42105663153250803765800904428, −1.35339021441263377182846020592, −1.34667682045832619217140641807, −0.953064942155305445359618452209, −0.923051770328367466503785915202, −0.815266083743101905152997298599, −0.75046854305491917465400561184, −0.62714670798349759059963930407, −0.61875544350948590153962278466, −0.58621212800487109111004127215, −0.53376113318604194884685354598, −0.52912085240392158444916125244, −0.13277571471093191317180458763,
0.13277571471093191317180458763, 0.52912085240392158444916125244, 0.53376113318604194884685354598, 0.58621212800487109111004127215, 0.61875544350948590153962278466, 0.62714670798349759059963930407, 0.75046854305491917465400561184, 0.815266083743101905152997298599, 0.923051770328367466503785915202, 0.953064942155305445359618452209, 1.34667682045832619217140641807, 1.35339021441263377182846020592, 1.42105663153250803765800904428, 1.44328432241407339075380971020, 1.47198583317822233863059576074, 1.53290341171665545339208318797, 1.54883400628448331681135671755, 1.57029505551627597010535868641, 1.79682071162444831017703393448, 1.94803418864404580734222265455, 1.97955967923853686672077331736, 2.04702207310873682330929801011, 2.05866325366553463023917601241, 2.16922431261332485128108121999, 2.26241433539667885574442666574
Plot not available for L-functions of degree greater than 10.
