| L(s) = 1 | − 1.27·2-s − 3-s − 0.370·4-s − 5-s + 1.27·6-s − 7-s + 3.02·8-s + 9-s + 1.27·10-s + 4.90·11-s + 0.370·12-s + 4.50·13-s + 1.27·14-s + 15-s − 3.12·16-s + 3.98·17-s − 1.27·18-s + 8.07·19-s + 0.370·20-s + 21-s − 6.25·22-s + 23-s − 3.02·24-s + 25-s − 5.75·26-s − 27-s + 0.370·28-s + ⋯ |
| L(s) = 1 | − 0.902·2-s − 0.577·3-s − 0.185·4-s − 0.447·5-s + 0.521·6-s − 0.377·7-s + 1.06·8-s + 0.333·9-s + 0.403·10-s + 1.47·11-s + 0.106·12-s + 1.24·13-s + 0.341·14-s + 0.258·15-s − 0.780·16-s + 0.966·17-s − 0.300·18-s + 1.85·19-s + 0.0827·20-s + 0.218·21-s − 1.33·22-s + 0.208·23-s − 0.617·24-s + 0.200·25-s − 1.12·26-s − 0.192·27-s + 0.0699·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8816031809\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8816031809\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 1.27T + 2T^{2} \) |
| 11 | \( 1 - 4.90T + 11T^{2} \) |
| 13 | \( 1 - 4.50T + 13T^{2} \) |
| 17 | \( 1 - 3.98T + 17T^{2} \) |
| 19 | \( 1 - 8.07T + 19T^{2} \) |
| 29 | \( 1 + 7.63T + 29T^{2} \) |
| 31 | \( 1 + 5.07T + 31T^{2} \) |
| 37 | \( 1 - 5.24T + 37T^{2} \) |
| 41 | \( 1 + 8.82T + 41T^{2} \) |
| 43 | \( 1 + 4.72T + 43T^{2} \) |
| 47 | \( 1 - 3.44T + 47T^{2} \) |
| 53 | \( 1 - 13.8T + 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 - 0.415T + 67T^{2} \) |
| 71 | \( 1 + 0.0137T + 71T^{2} \) |
| 73 | \( 1 - 5.23T + 73T^{2} \) |
| 79 | \( 1 - 2.49T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 - 2.83T + 89T^{2} \) |
| 97 | \( 1 + 16.9T + 97T^{2} \) |
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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
−9.133662435000231547106763101215, −8.284519622148877946916160866553, −7.43692929437623967758995506624, −6.89432566256988056550015820829, −5.85361897605639235864534884119, −5.14598395976961958506298022275, −3.86217022615093919348419445518, −3.55118449290751029380214282970, −1.50576794834911577385317012964, −0.808677558757411918142903402311,
0.808677558757411918142903402311, 1.50576794834911577385317012964, 3.55118449290751029380214282970, 3.86217022615093919348419445518, 5.14598395976961958506298022275, 5.85361897605639235864534884119, 6.89432566256988056550015820829, 7.43692929437623967758995506624, 8.284519622148877946916160866553, 9.133662435000231547106763101215
