| L(s) = 1 | − 1.13·3-s + 5-s + 2.44·7-s − 1.72·9-s + 3.35·11-s − 0.198·13-s − 1.13·15-s + 1.66·17-s − 2.14·19-s − 2.76·21-s − 7.53·23-s + 25-s + 5.33·27-s − 5.30·29-s − 3.84·31-s − 3.79·33-s + 2.44·35-s + 4.72·37-s + 0.225·39-s + 0.116·41-s − 10.5·43-s − 1.72·45-s − 1.58·47-s − 1.04·49-s − 1.87·51-s − 8.98·53-s + 3.35·55-s + ⋯ |
| L(s) = 1 | − 0.652·3-s + 0.447·5-s + 0.922·7-s − 0.573·9-s + 1.01·11-s − 0.0551·13-s − 0.291·15-s + 0.402·17-s − 0.491·19-s − 0.602·21-s − 1.57·23-s + 0.200·25-s + 1.02·27-s − 0.985·29-s − 0.691·31-s − 0.660·33-s + 0.412·35-s + 0.777·37-s + 0.0360·39-s + 0.0181·41-s − 1.61·43-s − 0.256·45-s − 0.230·47-s − 0.148·49-s − 0.263·51-s − 1.23·53-s + 0.452·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 151 | \( 1 + T \) |
| good | 3 | \( 1 + 1.13T + 3T^{2} \) |
| 7 | \( 1 - 2.44T + 7T^{2} \) |
| 11 | \( 1 - 3.35T + 11T^{2} \) |
| 13 | \( 1 + 0.198T + 13T^{2} \) |
| 17 | \( 1 - 1.66T + 17T^{2} \) |
| 19 | \( 1 + 2.14T + 19T^{2} \) |
| 23 | \( 1 + 7.53T + 23T^{2} \) |
| 29 | \( 1 + 5.30T + 29T^{2} \) |
| 31 | \( 1 + 3.84T + 31T^{2} \) |
| 37 | \( 1 - 4.72T + 37T^{2} \) |
| 41 | \( 1 - 0.116T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 + 1.58T + 47T^{2} \) |
| 53 | \( 1 + 8.98T + 53T^{2} \) |
| 59 | \( 1 + 4.35T + 59T^{2} \) |
| 61 | \( 1 + 5.10T + 61T^{2} \) |
| 67 | \( 1 + 2.74T + 67T^{2} \) |
| 71 | \( 1 - 5.00T + 71T^{2} \) |
| 73 | \( 1 + 3.52T + 73T^{2} \) |
| 79 | \( 1 - 17.1T + 79T^{2} \) |
| 83 | \( 1 - 5.89T + 83T^{2} \) |
| 89 | \( 1 + 10.7T + 89T^{2} \) |
| 97 | \( 1 + 8.51T + 97T^{2} \) |
| 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
−7.916641821665851397992489823653, −6.81301785222138965031309764771, −6.21330559003673573133719584373, −5.63884533963285313642586064572, −4.93875001096632055728814255324, −4.17151105377387771693280307566, −3.28894979636890304595798873552, −2.07063487157153659723220764878, −1.41276329652155994689393060864, 0,
1.41276329652155994689393060864, 2.07063487157153659723220764878, 3.28894979636890304595798873552, 4.17151105377387771693280307566, 4.93875001096632055728814255324, 5.63884533963285313642586064572, 6.21330559003673573133719584373, 6.81301785222138965031309764771, 7.916641821665851397992489823653
