| L(s) = 1 | + 2-s + 0.903·3-s + 4-s − 3.69·5-s + 0.903·6-s − 7-s + 8-s − 2.18·9-s − 3.69·10-s + 0.720·11-s + 0.903·12-s + 5.85·13-s − 14-s − 3.34·15-s + 16-s + 1.73·17-s − 2.18·18-s − 3.05·19-s − 3.69·20-s − 0.903·21-s + 0.720·22-s − 2.10·23-s + 0.903·24-s + 8.68·25-s + 5.85·26-s − 4.68·27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.521·3-s + 0.5·4-s − 1.65·5-s + 0.368·6-s − 0.377·7-s + 0.353·8-s − 0.728·9-s − 1.17·10-s + 0.217·11-s + 0.260·12-s + 1.62·13-s − 0.267·14-s − 0.862·15-s + 0.250·16-s + 0.420·17-s − 0.514·18-s − 0.701·19-s − 0.827·20-s − 0.197·21-s + 0.153·22-s − 0.437·23-s + 0.184·24-s + 1.73·25-s + 1.14·26-s − 0.901·27-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 | \( 1 - 0.903T + 3T^{2} \) |
| 5 | \( 1 + 3.69T + 5T^{2} \) |
| 11 | \( 1 - 0.720T + 11T^{2} \) |
| 13 | \( 1 - 5.85T + 13T^{2} \) |
| 17 | \( 1 - 1.73T + 17T^{2} \) |
| 19 | \( 1 + 3.05T + 19T^{2} \) |
| 23 | \( 1 + 2.10T + 23T^{2} \) |
| 29 | \( 1 + 1.23T + 29T^{2} \) |
| 31 | \( 1 - 4.25T + 31T^{2} \) |
| 37 | \( 1 - 0.0566T + 37T^{2} \) |
| 41 | \( 1 - 3.19T + 41T^{2} \) |
| 43 | \( 1 - 3.38T + 43T^{2} \) |
| 47 | \( 1 + 5.94T + 47T^{2} \) |
| 53 | \( 1 + 9.26T + 53T^{2} \) |
| 59 | \( 1 + 5.56T + 59T^{2} \) |
| 61 | \( 1 + 2.83T + 61T^{2} \) |
| 67 | \( 1 + 2.03T + 67T^{2} \) |
| 71 | \( 1 - 0.367T + 71T^{2} \) |
| 73 | \( 1 + 9.22T + 73T^{2} \) |
| 79 | \( 1 + 8.61T + 79T^{2} \) |
| 83 | \( 1 + 0.316T + 83T^{2} \) |
| 89 | \( 1 + 16.2T + 89T^{2} \) |
| 97 | \( 1 + 12.0T + 97T^{2} \) |
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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
−7.897894484600160656464291911317, −6.99746007068272069982272598231, −6.24846613536862394303765705582, −5.66944558542513381969292581006, −4.46123989342685363026309597797, −3.98813431967180569490534038857, −3.32860051953287200746697194841, −2.80296185059058046177341808498, −1.38811946671560530328694244619, 0,
1.38811946671560530328694244619, 2.80296185059058046177341808498, 3.32860051953287200746697194841, 3.98813431967180569490534038857, 4.46123989342685363026309597797, 5.66944558542513381969292581006, 6.24846613536862394303765705582, 6.99746007068272069982272598231, 7.897894484600160656464291911317
