| L(s) = 1 | − 2·2-s − 3-s + 2·4-s − 5-s + 2·6-s − 7-s − 2·9-s + 2·10-s − 2·12-s − 5·13-s + 2·14-s + 15-s − 4·16-s + 2·17-s + 4·18-s + 2·19-s − 2·20-s + 21-s − 2·23-s + 25-s + 10·26-s + 5·27-s − 2·28-s − 6·29-s − 2·30-s + 4·31-s + 8·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s − 0.447·5-s + 0.816·6-s − 0.377·7-s − 2/3·9-s + 0.632·10-s − 0.577·12-s − 1.38·13-s + 0.534·14-s + 0.258·15-s − 16-s + 0.485·17-s + 0.942·18-s + 0.458·19-s − 0.447·20-s + 0.218·21-s − 0.417·23-s + 1/5·25-s + 1.96·26-s + 0.962·27-s − 0.377·28-s − 1.11·29-s − 0.365·30-s + 0.718·31-s + 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 53 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−14.29976367356407, −13.86205888571684, −13.19217561150183, −12.58704928195067, −12.05799140341785, −11.73407020009385, −11.25756427969900, −10.74290445544482, −10.15358276369256, −9.900438313660370, −9.265719753861110, −8.944589560520615, −8.225081147775120, −7.788333515489317, −7.450390194619863, −6.811363575771418, −6.364201631816894, −5.644046712650866, −5.072833399529244, −4.622542318503932, −3.792803486140908, −3.066075461806231, −2.516715230852322, −1.743629891374841, −0.9725962016880258, 0, 0,
0.9725962016880258, 1.743629891374841, 2.516715230852322, 3.066075461806231, 3.792803486140908, 4.622542318503932, 5.072833399529244, 5.644046712650866, 6.364201631816894, 6.811363575771418, 7.450390194619863, 7.788333515489317, 8.225081147775120, 8.944589560520615, 9.265719753861110, 9.900438313660370, 10.15358276369256, 10.74290445544482, 11.25756427969900, 11.73407020009385, 12.05799140341785, 12.58704928195067, 13.19217561150183, 13.86205888571684, 14.29976367356407
