| L(s) = 1 | + 2·3-s − 3·4-s + 6·7-s + 6·9-s − 6·12-s + 5·16-s − 2·17-s − 12·19-s + 12·21-s − 10·23-s − 4·25-s + 12·27-s − 18·28-s − 8·29-s + 6·32-s − 18·36-s − 6·37-s − 12·41-s − 2·43-s + 10·48-s + 10·49-s − 4·51-s − 24·53-s − 24·57-s + 12·59-s − 28·61-s + 36·63-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 3/2·4-s + 2.26·7-s + 2·9-s − 1.73·12-s + 5/4·16-s − 0.485·17-s − 2.75·19-s + 2.61·21-s − 2.08·23-s − 4/5·25-s + 2.30·27-s − 3.40·28-s − 1.48·29-s + 1.06·32-s − 3·36-s − 0.986·37-s − 1.87·41-s − 0.304·43-s + 1.44·48-s + 10/7·49-s − 0.560·51-s − 3.29·53-s − 3.17·57-s + 1.56·59-s − 3.58·61-s + 4.53·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4646309796\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4646309796\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( ( 1 + T^{2} )^{4} \) |
| 13 | \( 1 \) |
| good | 2 | \( ( 1 - p T + p T^{2} - p T^{3} + T^{4} - p^{2} T^{5} + p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8} )( 1 + p T + 5 T^{2} + p^{3} T^{3} + 13 T^{4} + p^{4} T^{5} + 5 p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} ) \) |
| 3 | \( 1 - 2 T - 2 T^{2} + 4 T^{3} + T^{4} + 4 T^{5} + 10 T^{6} - 26 T^{7} - 20 T^{8} - 26 p T^{9} + 10 p^{2} T^{10} + 4 p^{3} T^{11} + p^{4} T^{12} + 4 p^{5} T^{13} - 2 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 7 | \( 1 - 6 T + 26 T^{2} - 12 p T^{3} + 229 T^{4} - 60 p T^{5} + 206 T^{6} + 954 T^{7} - 4268 T^{8} + 954 p T^{9} + 206 p^{2} T^{10} - 60 p^{4} T^{11} + 229 p^{4} T^{12} - 12 p^{6} T^{13} + 26 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 11 | \( 1 + 14 T^{2} + 97 T^{4} - 182 p T^{6} - 236 p^{2} T^{8} - 182 p^{3} T^{10} + 97 p^{4} T^{12} + 14 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( ( 1 - 14 T + 50 T^{2} + 220 T^{3} - 2129 T^{4} + 220 p T^{5} + 50 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )( 1 + 16 T + 128 T^{2} + 688 T^{3} + 3022 T^{4} + 688 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} ) \) |
| 19 | \( ( 1 + 6 T + 37 T^{2} + 150 T^{3} + 492 T^{4} + 150 p T^{5} + 37 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 + 10 T + 2 T^{2} - 108 T^{3} + 1013 T^{4} + 2804 T^{5} - 34298 T^{6} - 4266 p T^{7} + 8492 p T^{8} - 4266 p^{2} T^{9} - 34298 p^{2} T^{10} + 2804 p^{3} T^{11} + 1013 p^{4} T^{12} - 108 p^{5} T^{13} + 2 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) |
| 29 | \( 1 + 8 T - 34 T^{2} - 528 T^{3} + 353 T^{4} + 19264 T^{5} + 50686 T^{6} - 248280 T^{7} - 2118188 T^{8} - 248280 p T^{9} + 50686 p^{2} T^{10} + 19264 p^{3} T^{11} + 353 p^{4} T^{12} - 528 p^{5} T^{13} - 34 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) |
| 31 | \( ( 1 - 92 T^{2} + 3846 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( 1 + 6 T + 110 T^{2} + 588 T^{3} + 6289 T^{4} + 36600 T^{5} + 249746 T^{6} + 1600110 T^{7} + 8587516 T^{8} + 1600110 p T^{9} + 249746 p^{2} T^{10} + 36600 p^{3} T^{11} + 6289 p^{4} T^{12} + 588 p^{5} T^{13} + 110 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 41 | \( ( 1 + 6 T + 93 T^{2} + 486 T^{3} + 5372 T^{4} + 486 p T^{5} + 93 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 + 2 T - 150 T^{2} - 188 T^{3} + 13661 T^{4} + 10620 T^{5} - 862418 T^{6} - 185374 T^{7} + 42096180 T^{8} - 185374 p T^{9} - 862418 p^{2} T^{10} + 10620 p^{3} T^{11} + 13661 p^{4} T^{12} - 188 p^{5} T^{13} - 150 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 47 | \( 1 - 168 T^{2} + 17500 T^{4} - 1260696 T^{6} + 68079174 T^{8} - 1260696 p^{2} T^{10} + 17500 p^{4} T^{12} - 168 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( ( 1 + 12 T + 248 T^{2} + 36 p T^{3} + 20622 T^{4} + 36 p^{2} T^{5} + 248 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 - 12 T + 266 T^{2} - 2616 T^{3} + 37117 T^{4} - 331992 T^{5} + 3532226 T^{6} - 27597948 T^{7} + 240376924 T^{8} - 27597948 p T^{9} + 3532226 p^{2} T^{10} - 331992 p^{3} T^{11} + 37117 p^{4} T^{12} - 2616 p^{5} T^{13} + 266 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) |
| 61 | \( 1 + 28 T + 282 T^{2} + 1880 T^{3} + 26477 T^{4} + 344016 T^{5} + 2759698 T^{6} + 21027916 T^{7} + 175399068 T^{8} + 21027916 p T^{9} + 2759698 p^{2} T^{10} + 344016 p^{3} T^{11} + 26477 p^{4} T^{12} + 1880 p^{5} T^{13} + 282 p^{6} T^{14} + 28 p^{7} T^{15} + p^{8} T^{16} \) |
| 67 | \( 1 + 6 T + 166 T^{2} + 924 T^{3} + 11089 T^{4} + 78300 T^{5} + 988546 T^{6} + 7031886 T^{7} + 92397772 T^{8} + 7031886 p T^{9} + 988546 p^{2} T^{10} + 78300 p^{3} T^{11} + 11089 p^{4} T^{12} + 924 p^{5} T^{13} + 166 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 71 | \( 1 + 66 T^{2} - 5411 T^{4} + 1080 T^{5} - 5238 T^{6} - 2109672 T^{7} + 49128204 T^{8} - 2109672 p T^{9} - 5238 p^{2} T^{10} + 1080 p^{3} T^{11} - 5411 p^{4} T^{12} + 66 p^{6} T^{14} + p^{8} T^{16} \) |
| 73 | \( 1 - 352 T^{2} + 64540 T^{4} - 7783072 T^{6} + 668463622 T^{8} - 7783072 p^{2} T^{10} + 64540 p^{4} T^{12} - 352 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 + 8 T + 184 T^{2} + 1256 T^{3} + 21022 T^{4} + 1256 p T^{5} + 184 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 472 T^{2} + 104380 T^{4} - 14464552 T^{6} + 1407855142 T^{8} - 14464552 p^{2} T^{10} + 104380 p^{4} T^{12} - 472 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( 1 + 24 T + 338 T^{2} + 3504 T^{3} + 22477 T^{4} + 143640 T^{5} + 783482 T^{6} + 3203040 T^{7} + 47062828 T^{8} + 3203040 p T^{9} + 783482 p^{2} T^{10} + 143640 p^{3} T^{11} + 22477 p^{4} T^{12} + 3504 p^{5} T^{13} + 338 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \) |
| 97 | \( 1 - 30 T + 746 T^{2} - 13380 T^{3} + 217333 T^{4} - 2990232 T^{5} + 37765718 T^{6} - 423036774 T^{7} + 4390187620 T^{8} - 423036774 p T^{9} + 37765718 p^{2} T^{10} - 2990232 p^{3} T^{11} + 217333 p^{4} T^{12} - 13380 p^{5} T^{13} + 746 p^{6} T^{14} - 30 p^{7} T^{15} + p^{8} T^{16} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.46263040936414424395375586945, −4.24429743522534153261940345601, −4.06777535880419727042874366905, −3.97361344777388104785801931470, −3.91131876622579047076076689004, −3.88503630756856136380133426187, −3.81303934003622706954347466096, −3.35642803347030295311106146588, −3.28013464871661020532157837796, −3.23927522011344360915853430151, −3.09668163842090058156167352864, −2.90852282956867254796556878153, −2.87642520884560957235645193864, −2.36523694123015608002082040917, −2.19604894370553062122683139451, −2.02204097091666743621171475506, −1.98740678941239872205744928435, −1.88537350915967491844950530027, −1.81023599639487911648198610643, −1.70472303212183065326045678594, −1.42681899750208307142342359318, −1.26024434387821279599391438109, −0.78384325118826629933454676446, −0.52878220703345017852037466947, −0.088078047711832436623025571032,
0.088078047711832436623025571032, 0.52878220703345017852037466947, 0.78384325118826629933454676446, 1.26024434387821279599391438109, 1.42681899750208307142342359318, 1.70472303212183065326045678594, 1.81023599639487911648198610643, 1.88537350915967491844950530027, 1.98740678941239872205744928435, 2.02204097091666743621171475506, 2.19604894370553062122683139451, 2.36523694123015608002082040917, 2.87642520884560957235645193864, 2.90852282956867254796556878153, 3.09668163842090058156167352864, 3.23927522011344360915853430151, 3.28013464871661020532157837796, 3.35642803347030295311106146588, 3.81303934003622706954347466096, 3.88503630756856136380133426187, 3.91131876622579047076076689004, 3.97361344777388104785801931470, 4.06777535880419727042874366905, 4.24429743522534153261940345601, 4.46263040936414424395375586945
Plot not available for L-functions of degree greater than 10.
