| L(s) = 1 | + 2.02·2-s + 1.02·3-s + 2.08·4-s + 2.07·6-s + 1.36·7-s + 0.181·8-s − 1.94·9-s − 4.88·11-s + 2.14·12-s + 0.0386·13-s + 2.76·14-s − 3.81·16-s − 3.03·17-s − 3.93·18-s − 2.16·19-s + 1.40·21-s − 9.88·22-s − 0.0322·23-s + 0.186·24-s + 0.0781·26-s − 5.07·27-s + 2.85·28-s + 7.82·29-s − 8.10·31-s − 8.07·32-s − 5.01·33-s − 6.13·34-s + ⋯ |
| L(s) = 1 | + 1.42·2-s + 0.592·3-s + 1.04·4-s + 0.847·6-s + 0.517·7-s + 0.0641·8-s − 0.648·9-s − 1.47·11-s + 0.619·12-s + 0.0107·13-s + 0.739·14-s − 0.953·16-s − 0.735·17-s − 0.927·18-s − 0.495·19-s + 0.306·21-s − 2.10·22-s − 0.00672·23-s + 0.0380·24-s + 0.0153·26-s − 0.977·27-s + 0.540·28-s + 1.45·29-s − 1.45·31-s − 1.42·32-s − 0.873·33-s − 1.05·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 197 | \( 1 - T \) |
| good | 2 | \( 1 - 2.02T + 2T^{2} \) |
| 3 | \( 1 - 1.02T + 3T^{2} \) |
| 7 | \( 1 - 1.36T + 7T^{2} \) |
| 11 | \( 1 + 4.88T + 11T^{2} \) |
| 13 | \( 1 - 0.0386T + 13T^{2} \) |
| 17 | \( 1 + 3.03T + 17T^{2} \) |
| 19 | \( 1 + 2.16T + 19T^{2} \) |
| 23 | \( 1 + 0.0322T + 23T^{2} \) |
| 29 | \( 1 - 7.82T + 29T^{2} \) |
| 31 | \( 1 + 8.10T + 31T^{2} \) |
| 37 | \( 1 - 5.88T + 37T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 - 7.66T + 43T^{2} \) |
| 47 | \( 1 + 3.51T + 47T^{2} \) |
| 53 | \( 1 - 13.7T + 53T^{2} \) |
| 59 | \( 1 + 3.47T + 59T^{2} \) |
| 61 | \( 1 + 5.05T + 61T^{2} \) |
| 67 | \( 1 + 12.3T + 67T^{2} \) |
| 71 | \( 1 - 1.08T + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 - 2.67T + 79T^{2} \) |
| 83 | \( 1 + 2.27T + 83T^{2} \) |
| 89 | \( 1 + 2.18T + 89T^{2} \) |
| 97 | \( 1 - 17.1T + 97T^{2} \) |
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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
−7.85696411808209756114497225426, −7.09939682354225175886412064214, −6.16478045259582707198072496638, −5.55362948310809400320364159872, −4.86569429469528857977773519943, −4.27585709148997108348064157118, −3.27215500259753948214114972612, −2.68026521652995089052528476325, −1.99315449752863297989648972789, 0,
1.99315449752863297989648972789, 2.68026521652995089052528476325, 3.27215500259753948214114972612, 4.27585709148997108348064157118, 4.86569429469528857977773519943, 5.55362948310809400320364159872, 6.16478045259582707198072496638, 7.09939682354225175886412064214, 7.85696411808209756114497225426
