| L(s) = 1 | + 3·9-s − 12·11-s − 36·19-s + 36·29-s + 12·31-s + 6·41-s − 6·49-s + 6·59-s + 12·61-s + 64-s − 36·71-s − 12·79-s − 21·81-s − 36·99-s − 24·109-s + 108·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 48·169-s − 108·171-s + ⋯ |
| L(s) = 1 | + 9-s − 3.61·11-s − 8.25·19-s + 6.68·29-s + 2.15·31-s + 0.937·41-s − 6/7·49-s + 0.781·59-s + 1.53·61-s + 1/8·64-s − 4.27·71-s − 1.35·79-s − 7/3·81-s − 3.61·99-s − 2.29·109-s + 9.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 3.69·169-s − 8.25·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{24} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{24} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.137572841\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.137572841\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T^{6} + T^{12} \) |
| 5 | \( 1 \) |
| 19 | \( ( 1 + 18 T + 162 T^{2} + 883 T^{3} + 162 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| good | 3 | \( 1 - p T^{2} + 10 p T^{4} - 100 T^{6} + 131 p T^{8} - 491 p T^{10} + 3673 T^{12} - 491 p^{3} T^{14} + 131 p^{5} T^{16} - 100 p^{6} T^{18} + 10 p^{9} T^{20} - p^{11} T^{22} + p^{12} T^{24} \) |
| 7 | \( 1 + 6 T^{2} + 3 p T^{4} + 626 T^{6} + 1098 T^{8} + 1566 T^{10} + 209661 T^{12} + 1566 p^{2} T^{14} + 1098 p^{4} T^{16} + 626 p^{6} T^{18} + 3 p^{9} T^{20} + 6 p^{10} T^{22} + p^{12} T^{24} \) |
| 11 | \( ( 1 + 6 T - 10 T^{3} + 222 T^{4} + 42 T^{5} - 3181 T^{6} + 42 p T^{7} + 222 p^{2} T^{8} - 10 p^{3} T^{9} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 13 | \( 1 + 48 T^{2} + 1272 T^{4} + 26878 T^{6} + 498528 T^{8} + 7957764 T^{10} + 109835259 T^{12} + 7957764 p^{2} T^{14} + 498528 p^{4} T^{16} + 26878 p^{6} T^{18} + 1272 p^{8} T^{20} + 48 p^{10} T^{22} + p^{12} T^{24} \) |
| 17 | \( 1 - 72 T^{2} + 2304 T^{4} - 36578 T^{6} - 20088 T^{8} + 15903864 T^{10} - 398146461 T^{12} + 15903864 p^{2} T^{14} - 20088 p^{4} T^{16} - 36578 p^{6} T^{18} + 2304 p^{8} T^{20} - 72 p^{10} T^{22} + p^{12} T^{24} \) |
| 23 | \( 1 + 24 T^{2} - 216 T^{4} - 11514 T^{6} - 164592 T^{8} - 341868 T^{10} + 59040611 T^{12} - 341868 p^{2} T^{14} - 164592 p^{4} T^{16} - 11514 p^{6} T^{18} - 216 p^{8} T^{20} + 24 p^{10} T^{22} + p^{12} T^{24} \) |
| 29 | \( ( 1 - 18 T + 144 T^{2} - 20 p T^{3} + 1440 T^{4} - 7542 T^{5} + 56483 T^{6} - 7542 p T^{7} + 1440 p^{2} T^{8} - 20 p^{4} T^{9} + 144 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 31 | \( ( 1 - 6 T - 33 T^{2} + 346 T^{3} + 342 T^{4} - 6318 T^{5} + 21795 T^{6} - 6318 p T^{7} + 342 p^{2} T^{8} + 346 p^{3} T^{9} - 33 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 37 | \( ( 1 - 138 T^{2} + 10311 T^{4} - 467980 T^{6} + 10311 p^{2} T^{8} - 138 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 41 | \( ( 1 - 3 T + 6 T^{2} - 8 T^{3} - 111 T^{4} + 9711 T^{5} - 83311 T^{6} + 9711 p T^{7} - 111 p^{2} T^{8} - 8 p^{3} T^{9} + 6 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 43 | \( 1 - 48 T^{2} + 5244 T^{4} - 223802 T^{6} + 9743292 T^{8} - 518032764 T^{10} + 13239032139 T^{12} - 518032764 p^{2} T^{14} + 9743292 p^{4} T^{16} - 223802 p^{6} T^{18} + 5244 p^{8} T^{20} - 48 p^{10} T^{22} + p^{12} T^{24} \) |
| 47 | \( 1 + 156 T^{2} + 8136 T^{4} - 37374 T^{6} - 18749304 T^{8} - 435813936 T^{10} + 5244028307 T^{12} - 435813936 p^{2} T^{14} - 18749304 p^{4} T^{16} - 37374 p^{6} T^{18} + 8136 p^{8} T^{20} + 156 p^{10} T^{22} + p^{12} T^{24} \) |
| 53 | \( 1 + 24 T^{2} + 9144 T^{4} + 179106 T^{6} + 44216784 T^{8} + 673554372 T^{10} + 148633368011 T^{12} + 673554372 p^{2} T^{14} + 44216784 p^{4} T^{16} + 179106 p^{6} T^{18} + 9144 p^{8} T^{20} + 24 p^{10} T^{22} + p^{12} T^{24} \) |
| 59 | \( ( 1 - 3 T + 54 T^{2} - 378 T^{3} + 5823 T^{4} - 681 p T^{5} + 259255 T^{6} - 681 p^{2} T^{7} + 5823 p^{2} T^{8} - 378 p^{3} T^{9} + 54 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 61 | \( ( 1 - 6 T - 12 T^{2} - 586 T^{3} - 1008 T^{4} + 15732 T^{5} + 419787 T^{6} + 15732 p T^{7} - 1008 p^{2} T^{8} - 586 p^{3} T^{9} - 12 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 67 | \( 1 - 243 T^{2} + 37746 T^{4} - 4262848 T^{6} + 414234405 T^{8} - 34287692121 T^{10} + 2493013344825 T^{12} - 34287692121 p^{2} T^{14} + 414234405 p^{4} T^{16} - 4262848 p^{6} T^{18} + 37746 p^{8} T^{20} - 243 p^{10} T^{22} + p^{12} T^{24} \) |
| 71 | \( ( 1 + 18 T + 216 T^{2} + 2516 T^{3} + 26244 T^{4} + 221256 T^{5} + 1869317 T^{6} + 221256 p T^{7} + 26244 p^{2} T^{8} + 2516 p^{3} T^{9} + 216 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 73 | \( 1 + 342 T^{2} + 64503 T^{4} + 9149411 T^{6} + 1040827257 T^{8} + 97706269863 T^{10} + 7742659246590 T^{12} + 97706269863 p^{2} T^{14} + 1040827257 p^{4} T^{16} + 9149411 p^{6} T^{18} + 64503 p^{8} T^{20} + 342 p^{10} T^{22} + p^{12} T^{24} \) |
| 79 | \( ( 1 + 6 T - 48 T^{2} - 1430 T^{3} - 8820 T^{4} + 56934 T^{5} + 1359837 T^{6} + 56934 p T^{7} - 8820 p^{2} T^{8} - 1430 p^{3} T^{9} - 48 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 83 | \( 1 + 408 T^{2} + 91668 T^{4} + 14645398 T^{6} + 1841412492 T^{8} + 192460460724 T^{10} + 17188766926395 T^{12} + 192460460724 p^{2} T^{14} + 1841412492 p^{4} T^{16} + 14645398 p^{6} T^{18} + 91668 p^{8} T^{20} + 408 p^{10} T^{22} + p^{12} T^{24} \) |
| 89 | \( ( 1 + 36 T^{2} + 486 T^{3} - 1548 T^{4} + 34956 T^{5} + 586603 T^{6} + 34956 p T^{7} - 1548 p^{2} T^{8} + 486 p^{3} T^{9} + 36 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 97 | \( 1 - 3 T^{2} + 1158 T^{4} + 1655992 T^{6} - 14979447 T^{8} + 677893815 T^{10} + 1913630229513 T^{12} + 677893815 p^{2} T^{14} - 14979447 p^{4} T^{16} + 1655992 p^{6} T^{18} + 1158 p^{8} T^{20} - 3 p^{10} T^{22} + p^{12} T^{24} \) |
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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
−3.12134264800550263369224527904, −2.84197394023099063371516843951, −2.81661053996232694585438314755, −2.78086464352410073454881421983, −2.71460686469808871254820351602, −2.70524077417677992030758897301, −2.69850473104890684045769270328, −2.58418152274852687062925810556, −2.52653001240779259464581049072, −2.52199654723126744159355379671, −2.33632447480559448969649039650, −2.01733698992246024177908671257, −1.87448446011748354493572469758, −1.86851081302233195967276320604, −1.78452099702694801882761329097, −1.73873978338435342220101982829, −1.70395939253872750819905285909, −1.44889874202413374705389921408, −1.10722065392265758350151640006, −0.942215992823928429196010749589, −0.940264529552152716824966738260, −0.799693158013389479489915907756, −0.38455086387480233094104687602, −0.32999115519864626917321960981, −0.16169485516151435809572745482,
0.16169485516151435809572745482, 0.32999115519864626917321960981, 0.38455086387480233094104687602, 0.799693158013389479489915907756, 0.940264529552152716824966738260, 0.942215992823928429196010749589, 1.10722065392265758350151640006, 1.44889874202413374705389921408, 1.70395939253872750819905285909, 1.73873978338435342220101982829, 1.78452099702694801882761329097, 1.86851081302233195967276320604, 1.87448446011748354493572469758, 2.01733698992246024177908671257, 2.33632447480559448969649039650, 2.52199654723126744159355379671, 2.52653001240779259464581049072, 2.58418152274852687062925810556, 2.69850473104890684045769270328, 2.70524077417677992030758897301, 2.71460686469808871254820351602, 2.78086464352410073454881421983, 2.81661053996232694585438314755, 2.84197394023099063371516843951, 3.12134264800550263369224527904
Plot not available for L-functions of degree greater than 10.
