| L(s) = 1 | − 13.3·2-s − 163.·3-s − 333.·4-s + 2.18e3·6-s − 2.40e3·7-s + 1.12e4·8-s + 7.02e3·9-s − 9.01e4·11-s + 5.44e4·12-s + 3.19e3·13-s + 3.20e4·14-s + 1.98e4·16-s − 1.16e5·17-s − 9.38e4·18-s − 1.42e5·19-s + 3.92e5·21-s + 1.20e6·22-s − 1.27e6·23-s − 1.84e6·24-s − 4.27e4·26-s + 2.06e6·27-s + 8.00e5·28-s − 1.42e6·29-s + 9.67e6·31-s − 6.04e6·32-s + 1.47e7·33-s + 1.55e6·34-s + ⋯ |
| L(s) = 1 | − 0.590·2-s − 1.16·3-s − 0.651·4-s + 0.687·6-s − 0.377·7-s + 0.975·8-s + 0.356·9-s − 1.85·11-s + 0.758·12-s + 0.0310·13-s + 0.223·14-s + 0.0756·16-s − 0.338·17-s − 0.210·18-s − 0.250·19-s + 0.440·21-s + 1.09·22-s − 0.949·23-s − 1.13·24-s − 0.0183·26-s + 0.749·27-s + 0.246·28-s − 0.375·29-s + 1.88·31-s − 1.01·32-s + 2.16·33-s + 0.199·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + 2.40e3T \) |
| good | 2 | \( 1 + 13.3T + 512T^{2} \) |
| 3 | \( 1 + 163.T + 1.96e4T^{2} \) |
| 11 | \( 1 + 9.01e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 3.19e3T + 1.06e10T^{2} \) |
| 17 | \( 1 + 1.16e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 1.42e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.27e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.42e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 9.67e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 8.67e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.32e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.97e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.07e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 7.07e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 6.40e6T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.69e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.16e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.44e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.60e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.89e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 8.31e7T + 1.86e17T^{2} \) |
| 89 | \( 1 - 2.08e6T + 3.50e17T^{2} \) |
| 97 | \( 1 - 3.15e8T + 7.60e17T^{2} \) |
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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
−10.45020300800372408750883183008, −9.788530739794358081501145898745, −8.452508743579629956172730432645, −7.61320621028364699328295100302, −6.18725511061410202033328038541, −5.27484935332572328204873672872, −4.32909173481365273822846362323, −2.54732559978422719794758253914, −0.76588472468395388654175095457, 0,
0.76588472468395388654175095457, 2.54732559978422719794758253914, 4.32909173481365273822846362323, 5.27484935332572328204873672872, 6.18725511061410202033328038541, 7.61320621028364699328295100302, 8.452508743579629956172730432645, 9.788530739794358081501145898745, 10.45020300800372408750883183008
