| L(s) = 1 | + 15.4·2-s + 81·3-s − 272.·4-s + 2.70e3·5-s + 1.25e3·6-s + 6.16e3·7-s − 1.21e4·8-s + 6.56e3·9-s + 4.18e4·10-s − 3.28e4·11-s − 2.20e4·12-s + 5.11e4·13-s + 9.54e4·14-s + 2.19e5·15-s − 4.85e4·16-s + 3.15e5·17-s + 1.01e5·18-s − 3.44e5·19-s − 7.36e5·20-s + 4.99e5·21-s − 5.08e5·22-s + 1.91e6·23-s − 9.83e5·24-s + 5.36e6·25-s + 7.91e5·26-s + 5.31e5·27-s − 1.67e6·28-s + ⋯ |
| L(s) = 1 | + 0.684·2-s + 0.577·3-s − 0.531·4-s + 1.93·5-s + 0.394·6-s + 0.970·7-s − 1.04·8-s + 0.333·9-s + 1.32·10-s − 0.676·11-s − 0.307·12-s + 0.496·13-s + 0.664·14-s + 1.11·15-s − 0.185·16-s + 0.917·17-s + 0.228·18-s − 0.606·19-s − 1.02·20-s + 0.560·21-s − 0.463·22-s + 1.42·23-s − 0.605·24-s + 2.74·25-s + 0.339·26-s + 0.192·27-s − 0.516·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(5.603182771\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.603182771\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 81T \) |
| 59 | \( 1 + 1.21e7T \) |
| good | 2 | \( 1 - 15.4T + 512T^{2} \) |
| 5 | \( 1 - 2.70e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 6.16e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 3.28e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 5.11e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 3.15e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 3.44e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.91e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 9.38e4T + 1.45e13T^{2} \) |
| 31 | \( 1 + 1.74e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 5.42e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.76e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.64e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.13e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 2.10e7T + 3.29e15T^{2} \) |
| 61 | \( 1 + 2.14e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.80e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.63e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.32e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.93e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 6.43e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 5.28e7T + 3.50e17T^{2} \) |
| 97 | \( 1 + 6.12e8T + 7.60e17T^{2} \) |
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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
−10.83959823313234623571122847214, −9.918292284048216111788489845659, −9.035781914581144122914868277486, −8.214905310238428714815304528484, −6.57648005433134337275776916556, −5.40385807092824150645515504771, −4.91902339879200635043194216303, −3.29869982147003716217445803584, −2.19158322991846397667022540683, −1.12791342597850130003805843750,
1.12791342597850130003805843750, 2.19158322991846397667022540683, 3.29869982147003716217445803584, 4.91902339879200635043194216303, 5.40385807092824150645515504771, 6.57648005433134337275776916556, 8.214905310238428714815304528484, 9.035781914581144122914868277486, 9.918292284048216111788489845659, 10.83959823313234623571122847214
