| L(s) = 1 | + 2-s − 2.23·3-s + 4-s + 3.73·5-s − 2.23·6-s + 7-s + 8-s + 2.00·9-s + 3.73·10-s − 4.10·11-s − 2.23·12-s − 3.79·13-s + 14-s − 8.35·15-s + 16-s − 3.74·17-s + 2.00·18-s + 1.30·19-s + 3.73·20-s − 2.23·21-s − 4.10·22-s + 6.22·23-s − 2.23·24-s + 8.93·25-s − 3.79·26-s + 2.21·27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.29·3-s + 0.5·4-s + 1.66·5-s − 0.913·6-s + 0.377·7-s + 0.353·8-s + 0.669·9-s + 1.18·10-s − 1.23·11-s − 0.646·12-s − 1.05·13-s + 0.267·14-s − 2.15·15-s + 0.250·16-s − 0.909·17-s + 0.473·18-s + 0.300·19-s + 0.834·20-s − 0.488·21-s − 0.874·22-s + 1.29·23-s − 0.456·24-s + 1.78·25-s − 0.743·26-s + 0.426·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 + 2.23T + 3T^{2} \) |
| 5 | \( 1 - 3.73T + 5T^{2} \) |
| 11 | \( 1 + 4.10T + 11T^{2} \) |
| 13 | \( 1 + 3.79T + 13T^{2} \) |
| 17 | \( 1 + 3.74T + 17T^{2} \) |
| 19 | \( 1 - 1.30T + 19T^{2} \) |
| 23 | \( 1 - 6.22T + 23T^{2} \) |
| 29 | \( 1 + 8.57T + 29T^{2} \) |
| 31 | \( 1 + 4.88T + 31T^{2} \) |
| 37 | \( 1 - 0.368T + 37T^{2} \) |
| 41 | \( 1 - 5.37T + 41T^{2} \) |
| 43 | \( 1 + 7.45T + 43T^{2} \) |
| 47 | \( 1 + 8.59T + 47T^{2} \) |
| 53 | \( 1 + 9.89T + 53T^{2} \) |
| 59 | \( 1 - 14.9T + 59T^{2} \) |
| 61 | \( 1 + 13.3T + 61T^{2} \) |
| 67 | \( 1 + 2.08T + 67T^{2} \) |
| 71 | \( 1 - 7.69T + 71T^{2} \) |
| 73 | \( 1 - 4.51T + 73T^{2} \) |
| 79 | \( 1 + 16.6T + 79T^{2} \) |
| 83 | \( 1 - 4.38T + 83T^{2} \) |
| 89 | \( 1 + 9.37T + 89T^{2} \) |
| 97 | \( 1 + 18.3T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.29614563912535483480932448220, −6.83953103333418316034058632612, −6.02600094009126995572842260742, −5.39813548145343128922641534668, −5.16241775391009584854147729771, −4.54486585828948768958984397555, −3.05206413857996890119257367787, −2.30087352491124736150637138479, −1.51772634399885905534284500360, 0,
1.51772634399885905534284500360, 2.30087352491124736150637138479, 3.05206413857996890119257367787, 4.54486585828948768958984397555, 5.16241775391009584854147729771, 5.39813548145343128922641534668, 6.02600094009126995572842260742, 6.83953103333418316034058632612, 7.29614563912535483480932448220
