| L(s) = 1 | + 2-s − 4-s − 5-s + 7-s − 3·8-s − 3·9-s − 10-s − 6·11-s + 2·13-s + 14-s − 16-s − 2·17-s − 3·18-s − 8·19-s + 20-s − 6·22-s − 2·23-s + 25-s + 2·26-s − 28-s − 4·31-s + 5·32-s − 2·34-s − 35-s + 3·36-s + 2·37-s − 8·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.447·5-s + 0.377·7-s − 1.06·8-s − 9-s − 0.316·10-s − 1.80·11-s + 0.554·13-s + 0.267·14-s − 1/4·16-s − 0.485·17-s − 0.707·18-s − 1.83·19-s + 0.223·20-s − 1.27·22-s − 0.417·23-s + 1/5·25-s + 0.392·26-s − 0.188·28-s − 0.718·31-s + 0.883·32-s − 0.342·34-s − 0.169·35-s + 1/2·36-s + 0.328·37-s − 1.29·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 47 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.34096081779617, −13.56980553332526, −13.22031702716889, −13.11474004968326, −12.29342965363969, −11.95391657673173, −11.27642265993334, −10.86363663060558, −10.44622966296098, −9.828329228285263, −9.061902539855052, −8.471914816286704, −8.289259555757835, −7.936497999442689, −6.938581669422350, −6.461398890334746, −5.787363477460404, −5.273741983537432, −4.985295507035792, −4.237354866177774, −3.757795351513092, −3.148324218608396, −2.431403976995226, −2.009382041647812, −0.5665815072751018, 0,
0.5665815072751018, 2.009382041647812, 2.431403976995226, 3.148324218608396, 3.757795351513092, 4.237354866177774, 4.985295507035792, 5.273741983537432, 5.787363477460404, 6.461398890334746, 6.938581669422350, 7.936497999442689, 8.289259555757835, 8.471914816286704, 9.061902539855052, 9.828329228285263, 10.44622966296098, 10.86363663060558, 11.27642265993334, 11.95391657673173, 12.29342965363969, 13.11474004968326, 13.22031702716889, 13.56980553332526, 14.34096081779617
