| L(s) = 1 | + 2.43·2-s − 2.94·3-s + 3.94·4-s − 7.18·6-s + 3.82·7-s + 4.74·8-s + 5.67·9-s − 2.12·11-s − 11.6·12-s + 3.65·13-s + 9.31·14-s + 3.67·16-s + 3.04·17-s + 13.8·18-s − 11.2·21-s − 5.18·22-s + 4.81·23-s − 13.9·24-s + 8.91·26-s − 7.87·27-s + 15.0·28-s + 6.03·29-s + 3.57·31-s − 0.526·32-s + 6.25·33-s + 7.41·34-s + 22.3·36-s + ⋯ |
| L(s) = 1 | + 1.72·2-s − 1.70·3-s + 1.97·4-s − 2.93·6-s + 1.44·7-s + 1.67·8-s + 1.89·9-s − 0.640·11-s − 3.35·12-s + 1.01·13-s + 2.48·14-s + 0.918·16-s + 0.737·17-s + 3.26·18-s − 2.45·21-s − 1.10·22-s + 1.00·23-s − 2.85·24-s + 1.74·26-s − 1.51·27-s + 2.84·28-s + 1.12·29-s + 0.641·31-s − 0.0931·32-s + 1.08·33-s + 1.27·34-s + 3.73·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.664493396\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.664493396\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 2.43T + 2T^{2} \) |
| 3 | \( 1 + 2.94T + 3T^{2} \) |
| 7 | \( 1 - 3.82T + 7T^{2} \) |
| 11 | \( 1 + 2.12T + 11T^{2} \) |
| 13 | \( 1 - 3.65T + 13T^{2} \) |
| 17 | \( 1 - 3.04T + 17T^{2} \) |
| 23 | \( 1 - 4.81T + 23T^{2} \) |
| 29 | \( 1 - 6.03T + 29T^{2} \) |
| 31 | \( 1 - 3.57T + 31T^{2} \) |
| 37 | \( 1 + 3.93T + 37T^{2} \) |
| 41 | \( 1 + 7.60T + 41T^{2} \) |
| 43 | \( 1 - 5.60T + 43T^{2} \) |
| 47 | \( 1 - 8.41T + 47T^{2} \) |
| 53 | \( 1 + 4.80T + 53T^{2} \) |
| 59 | \( 1 + 5.13T + 59T^{2} \) |
| 61 | \( 1 + 13.5T + 61T^{2} \) |
| 67 | \( 1 - 5.38T + 67T^{2} \) |
| 71 | \( 1 + 0.123T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 + 4.25T + 79T^{2} \) |
| 83 | \( 1 + 4.39T + 83T^{2} \) |
| 89 | \( 1 + 0.0772T + 89T^{2} \) |
| 97 | \( 1 - 18.0T + 97T^{2} \) |
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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
−7.39793255403015554513978263230, −6.64611350360793692272676320576, −6.07251372509112376044536589245, −5.50419494785824083450154673177, −4.88689302759474788434071978619, −4.70785776280992132990129826363, −3.79764895521946885106927332339, −2.87116223854227584392366738266, −1.71439217026713930965330205453, −0.945636504506242838558107832417,
0.945636504506242838558107832417, 1.71439217026713930965330205453, 2.87116223854227584392366738266, 3.79764895521946885106927332339, 4.70785776280992132990129826363, 4.88689302759474788434071978619, 5.50419494785824083450154673177, 6.07251372509112376044536589245, 6.64611350360793692272676320576, 7.39793255403015554513978263230
