| L(s) = 1 | − 1.45·2-s + 2.46·3-s + 0.105·4-s − 3.68·5-s − 3.57·6-s − 0.804·7-s + 2.74·8-s + 3.06·9-s + 5.34·10-s + 2.63·11-s + 0.260·12-s − 1.95·13-s + 1.16·14-s − 9.06·15-s − 4.20·16-s + 0.337·17-s − 4.44·18-s + 2.70·19-s − 0.389·20-s − 1.98·21-s − 3.82·22-s − 2.03·23-s + 6.76·24-s + 8.55·25-s + 2.84·26-s + 0.161·27-s − 0.0851·28-s + ⋯ |
| L(s) = 1 | − 1.02·2-s + 1.42·3-s + 0.0529·4-s − 1.64·5-s − 1.45·6-s − 0.304·7-s + 0.971·8-s + 1.02·9-s + 1.68·10-s + 0.793·11-s + 0.0752·12-s − 0.543·13-s + 0.311·14-s − 2.34·15-s − 1.05·16-s + 0.0819·17-s − 1.04·18-s + 0.619·19-s − 0.0871·20-s − 0.432·21-s − 0.814·22-s − 0.423·23-s + 1.38·24-s + 1.71·25-s + 0.557·26-s + 0.0311·27-s − 0.0160·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 31 | \( 1 \) |
| good | 2 | \( 1 + 1.45T + 2T^{2} \) |
| 3 | \( 1 - 2.46T + 3T^{2} \) |
| 5 | \( 1 + 3.68T + 5T^{2} \) |
| 7 | \( 1 + 0.804T + 7T^{2} \) |
| 11 | \( 1 - 2.63T + 11T^{2} \) |
| 13 | \( 1 + 1.95T + 13T^{2} \) |
| 17 | \( 1 - 0.337T + 17T^{2} \) |
| 19 | \( 1 - 2.70T + 19T^{2} \) |
| 23 | \( 1 + 2.03T + 23T^{2} \) |
| 29 | \( 1 + 10.0T + 29T^{2} \) |
| 37 | \( 1 + 3.22T + 37T^{2} \) |
| 41 | \( 1 + 7.11T + 41T^{2} \) |
| 43 | \( 1 + 3.61T + 43T^{2} \) |
| 47 | \( 1 + 11.0T + 47T^{2} \) |
| 53 | \( 1 - 3.87T + 53T^{2} \) |
| 59 | \( 1 + 7.58T + 59T^{2} \) |
| 61 | \( 1 + 9.67T + 61T^{2} \) |
| 67 | \( 1 - 7.35T + 67T^{2} \) |
| 71 | \( 1 - 1.63T + 71T^{2} \) |
| 73 | \( 1 + 1.16T + 73T^{2} \) |
| 79 | \( 1 - 15.6T + 79T^{2} \) |
| 83 | \( 1 + 1.22T + 83T^{2} \) |
| 89 | \( 1 + 3.85T + 89T^{2} \) |
| 97 | \( 1 - 6.14T + 97T^{2} \) |
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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
−9.357943265546651584116239195596, −8.772975907148452095885471800987, −7.959276362299921926525068590674, −7.63187390481880775528521162542, −6.80296175266717166191328954189, −4.88518793965665532656005592592, −3.82887364840036564262779651918, −3.32385645375707530148963456761, −1.73246594256679659494983762842, 0,
1.73246594256679659494983762842, 3.32385645375707530148963456761, 3.82887364840036564262779651918, 4.88518793965665532656005592592, 6.80296175266717166191328954189, 7.63187390481880775528521162542, 7.959276362299921926525068590674, 8.772975907148452095885471800987, 9.357943265546651584116239195596
