| L(s) = 1 | + 2-s − 4-s − 4·5-s − 7-s − 3·8-s − 4·10-s − 2·13-s − 14-s − 16-s − 7·19-s + 4·20-s − 4·23-s + 11·25-s − 2·26-s + 28-s − 3·29-s + 7·31-s + 5·32-s + 4·35-s + 10·37-s − 7·38-s + 12·40-s − 2·43-s − 4·46-s − 3·47-s + 49-s + 11·50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.78·5-s − 0.377·7-s − 1.06·8-s − 1.26·10-s − 0.554·13-s − 0.267·14-s − 1/4·16-s − 1.60·19-s + 0.894·20-s − 0.834·23-s + 11/5·25-s − 0.392·26-s + 0.188·28-s − 0.557·29-s + 1.25·31-s + 0.883·32-s + 0.676·35-s + 1.64·37-s − 1.13·38-s + 1.89·40-s − 0.304·43-s − 0.589·46-s − 0.437·47-s + 1/7·49-s + 1.55·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18207 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.99196754286716, −15.20938564616589, −15.03745595240692, −14.59432416218976, −13.89300335147667, −13.13034110395076, −12.77062063938769, −12.31772439912574, −11.57173711982817, −11.52285704418950, −10.47521116828508, −10.01065379032745, −9.172156507841854, −8.635385272001833, −8.007768812575880, −7.700450132942973, −6.667198691084774, −6.357953588392160, −5.402112414077336, −4.686147509217060, −4.125308752691945, −3.876911113753477, −3.021999539820589, −2.333146905558623, −0.7238673612340793, 0,
0.7238673612340793, 2.333146905558623, 3.021999539820589, 3.876911113753477, 4.125308752691945, 4.686147509217060, 5.402112414077336, 6.357953588392160, 6.667198691084774, 7.700450132942973, 8.007768812575880, 8.635385272001833, 9.172156507841854, 10.01065379032745, 10.47521116828508, 11.52285704418950, 11.57173711982817, 12.31772439912574, 12.77062063938769, 13.13034110395076, 13.89300335147667, 14.59432416218976, 15.03745595240692, 15.20938564616589, 15.99196754286716
