| L(s) = 1 | + 0.152·5-s − 7-s − 0.385·11-s + 13-s − 7.43·17-s − 7.20·19-s − 2.90·23-s − 4.97·25-s + 5.20·29-s + 1.76·31-s − 0.152·35-s + 7.43·37-s − 7.05·41-s + 2.90·43-s + 3.59·47-s + 49-s + 10.9·53-s − 0.0587·55-s + 5.82·59-s + 12.5·61-s + 0.152·65-s + 9.80·67-s − 5.82·71-s + 3.09·73-s + 0.385·77-s + 12.6·79-s − 11.8·83-s + ⋯ |
| L(s) = 1 | + 0.0681·5-s − 0.377·7-s − 0.116·11-s + 0.277·13-s − 1.80·17-s − 1.65·19-s − 0.604·23-s − 0.995·25-s + 0.966·29-s + 0.317·31-s − 0.0257·35-s + 1.22·37-s − 1.10·41-s + 0.442·43-s + 0.523·47-s + 0.142·49-s + 1.50·53-s − 0.00791·55-s + 0.758·59-s + 1.60·61-s + 0.0189·65-s + 1.19·67-s − 0.691·71-s + 0.362·73-s + 0.0439·77-s + 1.42·79-s − 1.30·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.325253009\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.325253009\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 0.152T + 5T^{2} \) |
| 11 | \( 1 + 0.385T + 11T^{2} \) |
| 17 | \( 1 + 7.43T + 17T^{2} \) |
| 19 | \( 1 + 7.20T + 19T^{2} \) |
| 23 | \( 1 + 2.90T + 23T^{2} \) |
| 29 | \( 1 - 5.20T + 29T^{2} \) |
| 31 | \( 1 - 1.76T + 31T^{2} \) |
| 37 | \( 1 - 7.43T + 37T^{2} \) |
| 41 | \( 1 + 7.05T + 41T^{2} \) |
| 43 | \( 1 - 2.90T + 43T^{2} \) |
| 47 | \( 1 - 3.59T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 - 5.82T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 - 9.80T + 67T^{2} \) |
| 71 | \( 1 + 5.82T + 71T^{2} \) |
| 73 | \( 1 - 3.09T + 73T^{2} \) |
| 79 | \( 1 - 12.6T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 - 5.59T + 89T^{2} \) |
| 97 | \( 1 - 18.9T + 97T^{2} \) |
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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
−8.182509631859994266375241337196, −7.19993856528786689083962542269, −6.43070251191952680617075866576, −6.17168769215104357489302138287, −5.11338553377693740943539442895, −4.24240239874643498232540058496, −3.82939220350716653495954430040, −2.49361350297825209490912722151, −2.09112107651948929183086261254, −0.56540492065813242325640668957,
0.56540492065813242325640668957, 2.09112107651948929183086261254, 2.49361350297825209490912722151, 3.82939220350716653495954430040, 4.24240239874643498232540058496, 5.11338553377693740943539442895, 6.17168769215104357489302138287, 6.43070251191952680617075866576, 7.19993856528786689083962542269, 8.182509631859994266375241337196
