| L(s) = 1 | + 2·2-s + 3-s + 2·4-s + 2·6-s + 2·7-s − 2·9-s + 2·11-s + 2·12-s + 6·13-s + 4·14-s − 4·16-s + 7·17-s − 4·18-s − 5·19-s + 2·21-s + 4·22-s − 4·23-s + 12·26-s − 5·27-s + 4·28-s + 31-s − 8·32-s + 2·33-s + 14·34-s − 4·36-s + 7·37-s − 10·38-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s + 0.816·6-s + 0.755·7-s − 2/3·9-s + 0.603·11-s + 0.577·12-s + 1.66·13-s + 1.06·14-s − 16-s + 1.69·17-s − 0.942·18-s − 1.14·19-s + 0.436·21-s + 0.852·22-s − 0.834·23-s + 2.35·26-s − 0.962·27-s + 0.755·28-s + 0.179·31-s − 1.41·32-s + 0.348·33-s + 2.40·34-s − 2/3·36-s + 1.15·37-s − 1.62·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.965015555\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.965015555\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−10.58891803325594091684001275753, −9.315274066505842501962031785257, −8.452417840533593522893524117173, −7.85367458537759325623699515981, −6.30103705136471337181267013118, −5.90840703422548522617760576387, −4.76696248597564208150634186192, −3.79079605135641954029881998508, −3.13859783344881644507809684357, −1.71264203003942537105275855921,
1.71264203003942537105275855921, 3.13859783344881644507809684357, 3.79079605135641954029881998508, 4.76696248597564208150634186192, 5.90840703422548522617760576387, 6.30103705136471337181267013118, 7.85367458537759325623699515981, 8.452417840533593522893524117173, 9.315274066505842501962031785257, 10.58891803325594091684001275753
