| L(s) = 1 | + 2.67·2-s + 3-s + 5.15·4-s − 5-s + 2.67·6-s − 1.28·7-s + 8.44·8-s + 9-s − 2.67·10-s − 2.96·11-s + 5.15·12-s − 3.67·13-s − 3.44·14-s − 15-s + 12.2·16-s − 2.15·17-s + 2.67·18-s − 2.38·19-s − 5.15·20-s − 1.28·21-s − 7.92·22-s + 4.80·23-s + 8.44·24-s + 25-s − 9.83·26-s + 27-s − 6.63·28-s + ⋯ |
| L(s) = 1 | + 1.89·2-s + 0.577·3-s + 2.57·4-s − 0.447·5-s + 1.09·6-s − 0.486·7-s + 2.98·8-s + 0.333·9-s − 0.845·10-s − 0.893·11-s + 1.48·12-s − 1.01·13-s − 0.920·14-s − 0.258·15-s + 3.06·16-s − 0.522·17-s + 0.630·18-s − 0.547·19-s − 1.15·20-s − 0.280·21-s − 1.68·22-s + 1.00·23-s + 1.72·24-s + 0.200·25-s − 1.92·26-s + 0.192·27-s − 1.25·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.269014625\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.269014625\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 - 2.67T + 2T^{2} \) |
| 7 | \( 1 + 1.28T + 7T^{2} \) |
| 11 | \( 1 + 2.96T + 11T^{2} \) |
| 13 | \( 1 + 3.67T + 13T^{2} \) |
| 17 | \( 1 + 2.15T + 17T^{2} \) |
| 19 | \( 1 + 2.38T + 19T^{2} \) |
| 23 | \( 1 - 4.80T + 23T^{2} \) |
| 29 | \( 1 - 0.168T + 29T^{2} \) |
| 37 | \( 1 - 2.63T + 37T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 + 3.73T + 43T^{2} \) |
| 47 | \( 1 - 12.3T + 47T^{2} \) |
| 53 | \( 1 + 3.89T + 53T^{2} \) |
| 59 | \( 1 + 13.8T + 59T^{2} \) |
| 61 | \( 1 + 12.7T + 61T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 + 0.481T + 71T^{2} \) |
| 73 | \( 1 - 5.21T + 73T^{2} \) |
| 79 | \( 1 - 15.4T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 - 3.44T + 89T^{2} \) |
| 97 | \( 1 + 15.1T + 97T^{2} \) |
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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
−11.20676966813788259929592141194, −10.58454278173709757526296114642, −9.325087241850165629395985636598, −7.85809214604848448497196038107, −7.17997498153912541963279917808, −6.20872296595052740990885103658, −5.03027204279059755726977174379, −4.28186349879091723733665393629, −3.11840522466367847227150291557, −2.36609147327689149439071684279,
2.36609147327689149439071684279, 3.11840522466367847227150291557, 4.28186349879091723733665393629, 5.03027204279059755726977174379, 6.20872296595052740990885103658, 7.17997498153912541963279917808, 7.85809214604848448497196038107, 9.325087241850165629395985636598, 10.58454278173709757526296114642, 11.20676966813788259929592141194
