| L(s) = 1 | − 2.64·5-s − 7-s + 4.64·11-s − 5.29·13-s − 19-s + 6.64·23-s + 2.00·25-s + 6·29-s − 8.29·31-s + 2.64·35-s + 8.29·37-s + 3.35·41-s + 3.29·43-s + 0.708·47-s + 49-s − 12.2·55-s + 7.29·59-s + 14.5·61-s + 14.0·65-s + 9.29·67-s − 13.2·71-s + 11.2·73-s − 4.64·77-s − 0.708·79-s + 10·83-s + 1.93·89-s + 5.29·91-s + ⋯ |
| L(s) = 1 | − 1.18·5-s − 0.377·7-s + 1.40·11-s − 1.46·13-s − 0.229·19-s + 1.38·23-s + 0.400·25-s + 1.11·29-s − 1.48·31-s + 0.447·35-s + 1.36·37-s + 0.523·41-s + 0.501·43-s + 0.103·47-s + 0.142·49-s − 1.65·55-s + 0.949·59-s + 1.86·61-s + 1.73·65-s + 1.13·67-s − 1.56·71-s + 1.32·73-s − 0.529·77-s − 0.0797·79-s + 1.09·83-s + 0.205·89-s + 0.554·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.195091412\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.195091412\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.64T + 5T^{2} \) |
| 11 | \( 1 - 4.64T + 11T^{2} \) |
| 13 | \( 1 + 5.29T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 - 6.64T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 8.29T + 31T^{2} \) |
| 37 | \( 1 - 8.29T + 37T^{2} \) |
| 41 | \( 1 - 3.35T + 41T^{2} \) |
| 43 | \( 1 - 3.29T + 43T^{2} \) |
| 47 | \( 1 - 0.708T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 7.29T + 59T^{2} \) |
| 61 | \( 1 - 14.5T + 61T^{2} \) |
| 67 | \( 1 - 9.29T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 + 0.708T + 79T^{2} \) |
| 83 | \( 1 - 10T + 83T^{2} \) |
| 89 | \( 1 - 1.93T + 89T^{2} \) |
| 97 | \( 1 + 2.58T + 97T^{2} \) |
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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
−9.366184797415643190799092765940, −8.771355634458072893612904185037, −7.73109092804762682194749281308, −7.12659320088080962762624185554, −6.44300676070733511809596715081, −5.18722110553661988635645307966, −4.29057762335919791115583059679, −3.58752211783933304605986924838, −2.46859123881730287610229367449, −0.77158588562155336732208965900,
0.77158588562155336732208965900, 2.46859123881730287610229367449, 3.58752211783933304605986924838, 4.29057762335919791115583059679, 5.18722110553661988635645307966, 6.44300676070733511809596715081, 7.12659320088080962762624185554, 7.73109092804762682194749281308, 8.771355634458072893612904185037, 9.366184797415643190799092765940
