| L(s) = 1 | + 3.74·5-s + 7-s + 3.65·11-s − 5.54·13-s + 7.20·17-s − 3.30·19-s + 4.98·23-s + 9.05·25-s − 0.490·29-s − 3.89·31-s + 3.74·35-s + 7.89·37-s − 4.76·41-s − 1.60·43-s − 9.63·47-s + 49-s + 8.03·53-s + 13.6·55-s + 1.50·59-s − 2.08·61-s − 20.7·65-s + 3.40·67-s + 10.7·71-s + 9.83·73-s + 3.65·77-s + 3.72·79-s + 11.3·83-s + ⋯ |
| L(s) = 1 | + 1.67·5-s + 0.377·7-s + 1.10·11-s − 1.53·13-s + 1.74·17-s − 0.758·19-s + 1.04·23-s + 1.81·25-s − 0.0911·29-s − 0.699·31-s + 0.633·35-s + 1.29·37-s − 0.744·41-s − 0.244·43-s − 1.40·47-s + 0.142·49-s + 1.10·53-s + 1.84·55-s + 0.196·59-s − 0.267·61-s − 2.57·65-s + 0.415·67-s + 1.27·71-s + 1.15·73-s + 0.416·77-s + 0.419·79-s + 1.24·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.130131886\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.130131886\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 - 3.74T + 5T^{2} \) |
| 11 | \( 1 - 3.65T + 11T^{2} \) |
| 13 | \( 1 + 5.54T + 13T^{2} \) |
| 17 | \( 1 - 7.20T + 17T^{2} \) |
| 19 | \( 1 + 3.30T + 19T^{2} \) |
| 23 | \( 1 - 4.98T + 23T^{2} \) |
| 29 | \( 1 + 0.490T + 29T^{2} \) |
| 31 | \( 1 + 3.89T + 31T^{2} \) |
| 37 | \( 1 - 7.89T + 37T^{2} \) |
| 41 | \( 1 + 4.76T + 41T^{2} \) |
| 43 | \( 1 + 1.60T + 43T^{2} \) |
| 47 | \( 1 + 9.63T + 47T^{2} \) |
| 53 | \( 1 - 8.03T + 53T^{2} \) |
| 59 | \( 1 - 1.50T + 59T^{2} \) |
| 61 | \( 1 + 2.08T + 61T^{2} \) |
| 67 | \( 1 - 3.40T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 - 9.83T + 73T^{2} \) |
| 79 | \( 1 - 3.72T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 + 7.14T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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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
−8.408252146729659160304922224887, −7.52859872630155447451331086990, −6.76888137210618539790934070527, −6.17021796030230387820364666069, −5.26653600551740875743986194600, −4.95861154004653689483549481712, −3.73228775798425060329698857257, −2.69831879478649147507145627836, −1.91843635477693520873584456425, −1.06643081097423492147783894136,
1.06643081097423492147783894136, 1.91843635477693520873584456425, 2.69831879478649147507145627836, 3.73228775798425060329698857257, 4.95861154004653689483549481712, 5.26653600551740875743986194600, 6.17021796030230387820364666069, 6.76888137210618539790934070527, 7.52859872630155447451331086990, 8.408252146729659160304922224887
