| L(s) = 1 | − 85.0·2-s + 408.·3-s + 5.18e3·4-s − 7.33e3·5-s − 3.47e4·6-s + 1.68e4·7-s − 2.66e5·8-s − 1.04e4·9-s + 6.23e5·10-s − 6.16e5·11-s + 2.11e6·12-s − 1.73e6·13-s − 1.42e6·14-s − 2.99e6·15-s + 1.20e7·16-s + 2.25e6·17-s + 8.90e5·18-s − 4.46e6·19-s − 3.80e7·20-s + 6.86e6·21-s + 5.24e7·22-s − 3.07e7·23-s − 1.08e8·24-s + 4.90e6·25-s + 1.47e8·26-s − 7.65e7·27-s + 8.71e7·28-s + ⋯ |
| L(s) = 1 | − 1.87·2-s + 0.969·3-s + 2.53·4-s − 1.04·5-s − 1.82·6-s + 0.377·7-s − 2.87·8-s − 0.0591·9-s + 1.97·10-s − 1.15·11-s + 2.45·12-s − 1.29·13-s − 0.710·14-s − 1.01·15-s + 2.87·16-s + 0.386·17-s + 0.111·18-s − 0.413·19-s − 2.65·20-s + 0.366·21-s + 2.17·22-s − 0.997·23-s − 2.79·24-s + 0.100·25-s + 2.43·26-s − 1.02·27-s + 0.956·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 1.68e4T \) |
| good | 2 | \( 1 + 85.0T + 2.04e3T^{2} \) |
| 3 | \( 1 - 408.T + 1.77e5T^{2} \) |
| 5 | \( 1 + 7.33e3T + 4.88e7T^{2} \) |
| 11 | \( 1 + 6.16e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 1.73e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 2.25e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 4.46e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 3.07e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 1.42e8T + 1.22e16T^{2} \) |
| 31 | \( 1 - 7.19e7T + 2.54e16T^{2} \) |
| 37 | \( 1 - 6.71e7T + 1.77e17T^{2} \) |
| 41 | \( 1 + 1.76e7T + 5.50e17T^{2} \) |
| 43 | \( 1 - 3.80e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 1.92e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 1.67e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 7.61e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 4.21e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 6.85e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 1.56e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 2.70e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 2.62e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 2.20e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 2.81e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 1.07e11T + 7.15e21T^{2} \) |
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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
−19.16634940148811776685437207745, −17.69351713343921008779321530833, −16.06868145508675627531740066023, −14.87829284700560639909181318743, −11.82173007109350275080356632240, −10.11086410583036607039992074576, −8.358865591234548640992149568564, −7.62522232690598032193300036788, −2.53913838777535488923396680951, 0,
2.53913838777535488923396680951, 7.62522232690598032193300036788, 8.358865591234548640992149568564, 10.11086410583036607039992074576, 11.82173007109350275080356632240, 14.87829284700560639909181318743, 16.06868145508675627531740066023, 17.69351713343921008779321530833, 19.16634940148811776685437207745
