| L(s) = 1 | − 702.·2-s + 6.56e3·3-s + 3.62e5·4-s + 3.90e5·5-s − 4.60e6·6-s − 2.67e6·7-s − 1.62e8·8-s + 4.30e7·9-s − 2.74e8·10-s − 9.86e8·11-s + 2.37e9·12-s + 2.81e9·13-s + 1.87e9·14-s + 2.56e9·15-s + 6.66e10·16-s + 1.28e10·17-s − 3.02e10·18-s − 1.22e11·19-s + 1.41e11·20-s − 1.75e10·21-s + 6.92e11·22-s + 1.14e11·23-s − 1.06e12·24-s + 1.52e11·25-s − 1.97e12·26-s + 2.82e11·27-s − 9.69e11·28-s + ⋯ |
| L(s) = 1 | − 1.94·2-s + 0.577·3-s + 2.76·4-s + 0.447·5-s − 1.12·6-s − 0.175·7-s − 3.42·8-s + 0.333·9-s − 0.867·10-s − 1.38·11-s + 1.59·12-s + 0.957·13-s + 0.340·14-s + 0.258·15-s + 3.87·16-s + 0.446·17-s − 0.646·18-s − 1.65·19-s + 1.23·20-s − 0.101·21-s + 2.69·22-s + 0.305·23-s − 1.97·24-s + 0.200·25-s − 1.85·26-s + 0.192·27-s − 0.484·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 6.56e3T \) |
| 5 | \( 1 - 3.90e5T \) |
| good | 2 | \( 1 + 702.T + 1.31e5T^{2} \) |
| 7 | \( 1 + 2.67e6T + 2.32e14T^{2} \) |
| 11 | \( 1 + 9.86e8T + 5.05e17T^{2} \) |
| 13 | \( 1 - 2.81e9T + 8.65e18T^{2} \) |
| 17 | \( 1 - 1.28e10T + 8.27e20T^{2} \) |
| 19 | \( 1 + 1.22e11T + 5.48e21T^{2} \) |
| 23 | \( 1 - 1.14e11T + 1.41e23T^{2} \) |
| 29 | \( 1 + 4.90e12T + 7.25e24T^{2} \) |
| 31 | \( 1 - 7.64e12T + 2.25e25T^{2} \) |
| 37 | \( 1 - 3.18e12T + 4.56e26T^{2} \) |
| 41 | \( 1 + 5.53e12T + 2.61e27T^{2} \) |
| 43 | \( 1 - 2.42e13T + 5.87e27T^{2} \) |
| 47 | \( 1 + 5.82e13T + 2.66e28T^{2} \) |
| 53 | \( 1 + 6.45e14T + 2.05e29T^{2} \) |
| 59 | \( 1 + 1.04e12T + 1.27e30T^{2} \) |
| 61 | \( 1 + 1.56e15T + 2.24e30T^{2} \) |
| 67 | \( 1 - 1.31e15T + 1.10e31T^{2} \) |
| 71 | \( 1 + 7.09e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + 1.17e16T + 4.74e31T^{2} \) |
| 79 | \( 1 + 7.14e15T + 1.81e32T^{2} \) |
| 83 | \( 1 - 9.04e15T + 4.21e32T^{2} \) |
| 89 | \( 1 + 1.16e16T + 1.37e33T^{2} \) |
| 97 | \( 1 + 8.32e16T + 5.95e33T^{2} \) |
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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
−15.08974885145552067027826954698, −12.96179831693279076617456057667, −10.97650862413778942238206170504, −10.01220666376025558806813991229, −8.745350747949046446918439387358, −7.76089882839977728686029426365, −6.20556377686049608903008253241, −2.84017545143651075186639676318, −1.61711192957939357791759863108, 0,
1.61711192957939357791759863108, 2.84017545143651075186639676318, 6.20556377686049608903008253241, 7.76089882839977728686029426365, 8.745350747949046446918439387358, 10.01220666376025558806813991229, 10.97650862413778942238206170504, 12.96179831693279076617456057667, 15.08974885145552067027826954698
