| L(s) = 1 | − 5-s − 7-s + 5.65·11-s + 2·13-s + 3.65·17-s + 5.65·19-s − 5.65·23-s + 25-s − 3.65·29-s − 4·31-s + 35-s + 11.6·37-s − 2·41-s − 1.65·43-s − 2.34·47-s + 49-s + 3.65·53-s − 5.65·55-s − 4·59-s + 0.343·61-s − 2·65-s + 9.65·67-s − 7.31·71-s + 6·73-s − 5.65·77-s − 11.3·79-s − 4·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s + 1.70·11-s + 0.554·13-s + 0.886·17-s + 1.29·19-s − 1.17·23-s + 0.200·25-s − 0.679·29-s − 0.718·31-s + 0.169·35-s + 1.91·37-s − 0.312·41-s − 0.252·43-s − 0.341·47-s + 0.142·49-s + 0.502·53-s − 0.762·55-s − 0.520·59-s + 0.0439·61-s − 0.248·65-s + 1.17·67-s − 0.867·71-s + 0.702·73-s − 0.644·77-s − 1.27·79-s − 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.040728682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.040728682\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 - 5.65T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 3.65T + 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 + 5.65T + 23T^{2} \) |
| 29 | \( 1 + 3.65T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 11.6T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 1.65T + 43T^{2} \) |
| 47 | \( 1 + 2.34T + 47T^{2} \) |
| 53 | \( 1 - 3.65T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 0.343T + 61T^{2} \) |
| 67 | \( 1 - 9.65T + 67T^{2} \) |
| 71 | \( 1 + 7.31T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−8.144496655146578115068019607187, −7.55382896361391128098349748825, −6.80052950708203259944985651077, −6.07554849084156276582361082537, −5.48050491177818719325808977729, −4.29996073161548645706551260595, −3.73869707202321974767942413859, −3.10839597742653596381782942248, −1.72932186584518389511628970631, −0.826942759105745782965912745015,
0.826942759105745782965912745015, 1.72932186584518389511628970631, 3.10839597742653596381782942248, 3.73869707202321974767942413859, 4.29996073161548645706551260595, 5.48050491177818719325808977729, 6.07554849084156276582361082537, 6.80052950708203259944985651077, 7.55382896361391128098349748825, 8.144496655146578115068019607187
