| L(s) = 1 | − 4·2-s − 5·3-s + 4·4-s − 24·5-s + 20·6-s − 3·7-s + 2·8-s + 12·9-s + 96·10-s − 5·11-s − 20·12-s − 5·13-s + 12·14-s + 120·15-s − 16-s − 5·17-s − 48·18-s − 6·19-s − 96·20-s + 15·21-s + 20·22-s − 11·23-s − 10·24-s + 284·25-s + 20·26-s − 5·27-s − 12·28-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 2.88·3-s + 2·4-s − 10.7·5-s + 8.16·6-s − 1.13·7-s + 0.707·8-s + 4·9-s + 30.3·10-s − 1.50·11-s − 5.77·12-s − 1.38·13-s + 3.20·14-s + 30.9·15-s − 1/4·16-s − 1.21·17-s − 11.3·18-s − 1.37·19-s − 21.4·20-s + 3.27·21-s + 4.26·22-s − 2.29·23-s − 2.04·24-s + 56.7·25-s + 3.92·26-s − 0.962·27-s − 2.26·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + 5 T + 12 T^{2} - 5 T^{3} - 181 T^{4} - 5 p T^{5} + 12 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | \( ( 1 - T - 39 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| good | 2 | \( ( 1 - T + p^{2} T^{2} - p T^{3} + 9 T^{4} - p^{2} T^{5} + p^{4} T^{6} - p^{3} T^{7} + p^{4} T^{8} )( 1 + 5 T + 13 T^{2} + 25 T^{3} + 39 T^{4} + 25 p T^{5} + 13 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} ) \) |
| 3 | \( 1 + 5 T + 13 T^{2} + 10 T^{3} - 10 p T^{4} - 115 T^{5} - 122 T^{6} + 100 T^{7} + 499 T^{8} + 100 p T^{9} - 122 p^{2} T^{10} - 115 p^{3} T^{11} - 10 p^{5} T^{12} + 10 p^{5} T^{13} + 13 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) |
| 5 | \( ( 1 + 3 T + p T^{2} )^{8} \) |
| 7 | \( 1 + 3 T + p T^{2} + 36 T^{3} + 108 T^{4} + 219 T^{5} + 122 p T^{6} + 2628 T^{7} + 5483 T^{8} + 2628 p T^{9} + 122 p^{3} T^{10} + 219 p^{3} T^{11} + 108 p^{4} T^{12} + 36 p^{5} T^{13} + p^{7} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) |
| 11 | \( 1 + 5 T + 21 T^{2} - 30 T^{3} - 310 T^{4} - 1635 T^{5} - 586 T^{6} + 11900 T^{7} + 83499 T^{8} + 11900 p T^{9} - 586 p^{2} T^{10} - 1635 p^{3} T^{11} - 310 p^{4} T^{12} - 30 p^{5} T^{13} + 21 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) |
| 17 | \( 1 + 5 T + 32 T^{2} + 135 T^{3} + 670 T^{4} + 1110 T^{5} + 9032 T^{6} + 640 p T^{7} + 63559 T^{8} + 640 p^{2} T^{9} + 9032 p^{2} T^{10} + 1110 p^{3} T^{11} + 670 p^{4} T^{12} + 135 p^{5} T^{13} + 32 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) |
| 19 | \( 1 + 6 T + p T^{2} + 126 T^{3} + 756 T^{4} + 4308 T^{5} + 1079 p T^{6} + 90468 T^{7} + 412487 T^{8} + 90468 p T^{9} + 1079 p^{3} T^{10} + 4308 p^{3} T^{11} + 756 p^{4} T^{12} + 126 p^{5} T^{13} + p^{7} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 23 | \( 1 + 11 T + 93 T^{2} + 798 T^{3} + 5738 T^{4} + 37203 T^{5} + 214886 T^{6} + 1165784 T^{7} + 5972343 T^{8} + 1165784 p T^{9} + 214886 p^{2} T^{10} + 37203 p^{3} T^{11} + 5738 p^{4} T^{12} + 798 p^{5} T^{13} + 93 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) |
| 29 | \( 1 + 7 T + 54 T^{2} + 411 T^{3} + 2762 T^{4} + 11112 T^{5} + 80708 T^{6} + 338866 T^{7} + 1602651 T^{8} + 338866 p T^{9} + 80708 p^{2} T^{10} + 11112 p^{3} T^{11} + 2762 p^{4} T^{12} + 411 p^{5} T^{13} + 54 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) |
| 37 | \( ( 1 - 5 T - 24 T^{2} + 125 T^{3} + 107 T^{4} + 125 p T^{5} - 24 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( 1 - 17 T + 146 T^{2} - 153 T^{3} - 7514 T^{4} + 82824 T^{5} - 245596 T^{6} - 1697654 T^{7} + 23405647 T^{8} - 1697654 p T^{9} - 245596 p^{2} T^{10} + 82824 p^{3} T^{11} - 7514 p^{4} T^{12} - 153 p^{5} T^{13} + 146 p^{6} T^{14} - 17 p^{7} T^{15} + p^{8} T^{16} \) |
| 43 | \( 1 + 12 T + 123 T^{2} + 340 T^{3} + 240 T^{4} - 20312 T^{5} + 2023 p T^{6} + 1467420 T^{7} + 19111999 T^{8} + 1467420 p T^{9} + 2023 p^{3} T^{10} - 20312 p^{3} T^{11} + 240 p^{4} T^{12} + 340 p^{5} T^{13} + 123 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) |
| 47 | \( ( 1 + 14 T + 29 T^{2} - 352 T^{3} - 2851 T^{4} - 352 p T^{5} + 29 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 4 T - 47 T^{2} + 160 T^{3} + 2121 T^{4} + 160 p T^{5} - 47 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 - 4 T - 21 T^{2} - 144 T^{3} + 836 T^{4} + 4008 T^{5} + 237701 T^{6} - 1329022 T^{7} - 10035333 T^{8} - 1329022 p T^{9} + 237701 p^{2} T^{10} + 4008 p^{3} T^{11} + 836 p^{4} T^{12} - 144 p^{5} T^{13} - 21 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 61 | \( ( 1 + 3 T - 14 T^{2} - 297 T^{3} - 3741 T^{4} - 297 p T^{5} - 14 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + 20 T + 171 T^{2} + 1900 T^{3} + 21152 T^{4} + 1900 p T^{5} + 171 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( 1 + 19 T + 291 T^{2} + 4008 T^{3} + 50192 T^{4} + 565539 T^{5} + 5694542 T^{6} + 53394988 T^{7} + 475063971 T^{8} + 53394988 p T^{9} + 5694542 p^{2} T^{10} + 565539 p^{3} T^{11} + 50192 p^{4} T^{12} + 4008 p^{5} T^{13} + 291 p^{6} T^{14} + 19 p^{7} T^{15} + p^{8} T^{16} \) |
| 73 | \( ( 1 + 22 T + 171 T^{2} + 1076 T^{3} + 10229 T^{4} + 1076 p T^{5} + 171 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 27 T + 470 T^{2} + 6117 T^{3} + 61789 T^{4} + 6117 p T^{5} + 470 p^{2} T^{6} + 27 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 23 T + 2 p T^{2} + 709 T^{3} + 6369 T^{4} + 709 p T^{5} + 2 p^{3} T^{6} + 23 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 89 | \( 1 - 9 T + 134 T^{2} + 51 T^{3} - 1674 T^{4} + 122868 T^{5} - 3196 p T^{6} + 806238 T^{7} + 66134807 T^{8} + 806238 p T^{9} - 3196 p^{3} T^{10} + 122868 p^{3} T^{11} - 1674 p^{4} T^{12} + 51 p^{5} T^{13} + 134 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) |
| 97 | \( 1 + 3 T + 102 T^{2} - 89 T^{3} - 282 T^{4} - 55676 T^{5} - 8828 p T^{6} - 2476602 T^{7} - 69420377 T^{8} - 2476602 p T^{9} - 8828 p^{3} T^{10} - 55676 p^{3} T^{11} - 282 p^{4} T^{12} - 89 p^{5} T^{13} + 102 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) |
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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
−5.53409627016609236461521871133, −5.34062808333364824405762146828, −5.26641125180554938714487943133, −4.86584692609589067645412999019, −4.83409769174376064458904890213, −4.59265405928374448558763414562, −4.44989847038934304305474795707, −4.44196001463483269870784846513, −4.41525181195799485634430434858, −4.25325330144221261186770234007, −4.18331137187489442870791741087, −4.10995565319241203093750224642, −3.96750977776426031706133739143, −3.79941532502813984928709922320, −3.62585364725276172765604642927, −3.59110727956735698517767464890, −3.12012843077606874288111467237, −2.96884146657611589151881090630, −2.94947077268940667191831812358, −2.80065530940836855957605674856, −2.74594843916206594532864652849, −2.51363062021581280999123310090, −1.36258020033197012993233889458, −1.25837349847692517321067560713, −1.19367856040792203542913487642, 0, 0, 0, 0, 0, 0, 0, 0,
1.19367856040792203542913487642, 1.25837349847692517321067560713, 1.36258020033197012993233889458, 2.51363062021581280999123310090, 2.74594843916206594532864652849, 2.80065530940836855957605674856, 2.94947077268940667191831812358, 2.96884146657611589151881090630, 3.12012843077606874288111467237, 3.59110727956735698517767464890, 3.62585364725276172765604642927, 3.79941532502813984928709922320, 3.96750977776426031706133739143, 4.10995565319241203093750224642, 4.18331137187489442870791741087, 4.25325330144221261186770234007, 4.41525181195799485634430434858, 4.44196001463483269870784846513, 4.44989847038934304305474795707, 4.59265405928374448558763414562, 4.83409769174376064458904890213, 4.86584692609589067645412999019, 5.26641125180554938714487943133, 5.34062808333364824405762146828, 5.53409627016609236461521871133
Plot not available for L-functions of degree greater than 10.
