| L(s) = 1 | + 9.66e3·2-s − 1.07e6·3-s + 5.98e7·4-s − 2.44e8·5-s − 1.03e10·6-s + 1.87e10·7-s + 2.54e11·8-s + 3.01e11·9-s − 2.36e12·10-s − 1.84e13·11-s − 6.41e13·12-s − 1.08e14·13-s + 1.81e14·14-s + 2.61e14·15-s + 4.51e14·16-s − 2.86e15·17-s + 2.91e15·18-s + 1.17e16·19-s − 1.46e16·20-s − 2.01e16·21-s − 1.78e17·22-s + 7.00e16·23-s − 2.72e17·24-s + 5.96e16·25-s − 1.04e18·26-s + 5.84e17·27-s + 1.12e18·28-s + ⋯ |
| L(s) = 1 | + 1.66·2-s − 1.16·3-s + 1.78·4-s − 0.447·5-s − 1.94·6-s + 0.513·7-s + 1.30·8-s + 0.356·9-s − 0.746·10-s − 1.77·11-s − 2.07·12-s − 1.29·13-s + 0.856·14-s + 0.520·15-s + 0.400·16-s − 1.19·17-s + 0.594·18-s + 1.22·19-s − 0.798·20-s − 0.597·21-s − 2.95·22-s + 0.666·23-s − 1.52·24-s + 0.199·25-s − 2.15·26-s + 0.749·27-s + 0.916·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{27}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 2.44e8T \) |
| good | 2 | \( 1 - 9.66e3T + 3.35e7T^{2} \) |
| 3 | \( 1 + 1.07e6T + 8.47e11T^{2} \) |
| 7 | \( 1 - 1.87e10T + 1.34e21T^{2} \) |
| 11 | \( 1 + 1.84e13T + 1.08e26T^{2} \) |
| 13 | \( 1 + 1.08e14T + 7.05e27T^{2} \) |
| 17 | \( 1 + 2.86e15T + 5.77e30T^{2} \) |
| 19 | \( 1 - 1.17e16T + 9.30e31T^{2} \) |
| 23 | \( 1 - 7.00e16T + 1.10e34T^{2} \) |
| 29 | \( 1 + 7.10e17T + 3.63e36T^{2} \) |
| 31 | \( 1 + 2.41e18T + 1.92e37T^{2} \) |
| 37 | \( 1 - 1.09e19T + 1.60e39T^{2} \) |
| 41 | \( 1 - 2.12e20T + 2.08e40T^{2} \) |
| 43 | \( 1 + 5.14e20T + 6.86e40T^{2} \) |
| 47 | \( 1 - 9.21e20T + 6.34e41T^{2} \) |
| 53 | \( 1 - 4.27e21T + 1.27e43T^{2} \) |
| 59 | \( 1 - 4.93e21T + 1.86e44T^{2} \) |
| 61 | \( 1 + 2.30e20T + 4.29e44T^{2} \) |
| 67 | \( 1 + 5.89e22T + 4.48e45T^{2} \) |
| 71 | \( 1 + 1.62e23T + 1.91e46T^{2} \) |
| 73 | \( 1 + 3.59e22T + 3.82e46T^{2} \) |
| 79 | \( 1 + 1.76e23T + 2.75e47T^{2} \) |
| 83 | \( 1 - 6.94e23T + 9.48e47T^{2} \) |
| 89 | \( 1 - 1.06e24T + 5.42e48T^{2} \) |
| 97 | \( 1 - 6.40e23T + 4.66e49T^{2} \) |
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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
−16.18103903094293705675584228653, −14.95800704914318920576287659026, −13.18714412605120881379626224211, −11.92393074336226593211818009887, −10.88644556378378328873323917659, −7.27360139831276505213790813502, −5.45442372914748486355849666013, −4.73862563336786256498539309422, −2.65359274801333924497957295124, 0,
2.65359274801333924497957295124, 4.73862563336786256498539309422, 5.45442372914748486355849666013, 7.27360139831276505213790813502, 10.88644556378378328873323917659, 11.92393074336226593211818009887, 13.18714412605120881379626224211, 14.95800704914318920576287659026, 16.18103903094293705675584228653
