| L(s) = 1 | + 1.49·2-s − 0.0947·3-s + 0.236·4-s − 0.141·6-s − 4.82·7-s − 2.63·8-s − 2.99·9-s − 1.06·11-s − 0.0224·12-s − 7.21·14-s − 4.41·16-s − 3.55·17-s − 4.47·18-s + 5.73·19-s + 0.457·21-s − 1.59·22-s + 7.08·23-s + 0.249·24-s + 0.567·27-s − 1.14·28-s + 1.47·29-s − 1.46·31-s − 1.33·32-s + 0.101·33-s − 5.32·34-s − 0.707·36-s + 0.0253·37-s + ⋯ |
| L(s) = 1 | + 1.05·2-s − 0.0547·3-s + 0.118·4-s − 0.0578·6-s − 1.82·7-s − 0.932·8-s − 0.997·9-s − 0.322·11-s − 0.00647·12-s − 1.92·14-s − 1.10·16-s − 0.863·17-s − 1.05·18-s + 1.31·19-s + 0.0998·21-s − 0.340·22-s + 1.47·23-s + 0.0510·24-s + 0.109·27-s − 0.215·28-s + 0.273·29-s − 0.262·31-s − 0.235·32-s + 0.0176·33-s − 0.912·34-s − 0.117·36-s + 0.00417·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.309576634\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.309576634\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.49T + 2T^{2} \) |
| 3 | \( 1 + 0.0947T + 3T^{2} \) |
| 7 | \( 1 + 4.82T + 7T^{2} \) |
| 11 | \( 1 + 1.06T + 11T^{2} \) |
| 17 | \( 1 + 3.55T + 17T^{2} \) |
| 19 | \( 1 - 5.73T + 19T^{2} \) |
| 23 | \( 1 - 7.08T + 23T^{2} \) |
| 29 | \( 1 - 1.47T + 29T^{2} \) |
| 31 | \( 1 + 1.46T + 31T^{2} \) |
| 37 | \( 1 - 0.0253T + 37T^{2} \) |
| 41 | \( 1 - 0.267T + 41T^{2} \) |
| 43 | \( 1 + 3.55T + 43T^{2} \) |
| 47 | \( 1 + 6.51T + 47T^{2} \) |
| 53 | \( 1 + 0.991T + 53T^{2} \) |
| 59 | \( 1 - 8.72T + 59T^{2} \) |
| 61 | \( 1 - 6.33T + 61T^{2} \) |
| 67 | \( 1 + 5.17T + 67T^{2} \) |
| 71 | \( 1 + 7.76T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 - 8.78T + 79T^{2} \) |
| 83 | \( 1 - 0.725T + 83T^{2} \) |
| 89 | \( 1 - 13.5T + 89T^{2} \) |
| 97 | \( 1 - 3.43T + 97T^{2} \) |
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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
−8.650261659655597102594982663960, −7.39241028749774297801459400728, −6.64621538943561153430898291545, −6.11121323628171498725595707375, −5.38464026816075382585926093840, −4.77289239589637991497589320890, −3.60075717165485199145453344471, −3.16091299860889454942666357025, −2.55044516287328545730059277893, −0.51913145495065955529476892799,
0.51913145495065955529476892799, 2.55044516287328545730059277893, 3.16091299860889454942666357025, 3.60075717165485199145453344471, 4.77289239589637991497589320890, 5.38464026816075382585926093840, 6.11121323628171498725595707375, 6.64621538943561153430898291545, 7.39241028749774297801459400728, 8.650261659655597102594982663960
