| L(s) = 1 | − 2.44·2-s + 2.35·3-s + 3.98·4-s − 2.67·5-s − 5.76·6-s − 0.955·7-s − 4.86·8-s + 2.54·9-s + 6.54·10-s − 0.164·11-s + 9.39·12-s + 2.33·14-s − 6.30·15-s + 3.92·16-s − 0.846·17-s − 6.23·18-s + 4.61·19-s − 10.6·20-s − 2.25·21-s + 0.403·22-s − 3.55·23-s − 11.4·24-s + 2.15·25-s − 1.06·27-s − 3.81·28-s + 5.07·29-s + 15.4·30-s + ⋯ |
| L(s) = 1 | − 1.73·2-s + 1.35·3-s + 1.99·4-s − 1.19·5-s − 2.35·6-s − 0.361·7-s − 1.71·8-s + 0.849·9-s + 2.07·10-s − 0.0496·11-s + 2.71·12-s + 0.624·14-s − 1.62·15-s + 0.981·16-s − 0.205·17-s − 1.46·18-s + 1.05·19-s − 2.38·20-s − 0.491·21-s + 0.0859·22-s − 0.740·23-s − 2.33·24-s + 0.431·25-s − 0.205·27-s − 0.720·28-s + 0.942·29-s + 2.81·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 2.44T + 2T^{2} \) |
| 3 | \( 1 - 2.35T + 3T^{2} \) |
| 5 | \( 1 + 2.67T + 5T^{2} \) |
| 7 | \( 1 + 0.955T + 7T^{2} \) |
| 11 | \( 1 + 0.164T + 11T^{2} \) |
| 17 | \( 1 + 0.846T + 17T^{2} \) |
| 19 | \( 1 - 4.61T + 19T^{2} \) |
| 23 | \( 1 + 3.55T + 23T^{2} \) |
| 29 | \( 1 - 5.07T + 29T^{2} \) |
| 37 | \( 1 - 7.21T + 37T^{2} \) |
| 41 | \( 1 + 0.599T + 41T^{2} \) |
| 43 | \( 1 - 4.14T + 43T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 - 9.33T + 53T^{2} \) |
| 59 | \( 1 + 7.75T + 59T^{2} \) |
| 61 | \( 1 + 6.21T + 61T^{2} \) |
| 67 | \( 1 - 0.784T + 67T^{2} \) |
| 71 | \( 1 + 2.65T + 71T^{2} \) |
| 73 | \( 1 + 0.941T + 73T^{2} \) |
| 79 | \( 1 - 10.0T + 79T^{2} \) |
| 83 | \( 1 - 7.61T + 83T^{2} \) |
| 89 | \( 1 - 17.3T + 89T^{2} \) |
| 97 | \( 1 + 6.09T + 97T^{2} \) |
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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
−7.963015790617399196391904899767, −7.67624339404003582840862779277, −6.90527141878458281325120318808, −6.12117248973135536372865487306, −4.68298623237300544712467598007, −3.71869667333641733693633954089, −3.03493746093008949245990172116, −2.30494045029715226801806104331, −1.18408596888953319598875137964, 0,
1.18408596888953319598875137964, 2.30494045029715226801806104331, 3.03493746093008949245990172116, 3.71869667333641733693633954089, 4.68298623237300544712467598007, 6.12117248973135536372865487306, 6.90527141878458281325120318808, 7.67624339404003582840862779277, 7.963015790617399196391904899767
