| L(s) = 1 | + 10·2-s + 14·3-s + 68·4-s + 140·6-s + 49·7-s + 360·8-s − 47·9-s + 232·11-s + 952·12-s + 140·13-s + 490·14-s + 1.42e3·16-s + 1.72e3·17-s − 470·18-s − 98·19-s + 686·21-s + 2.32e3·22-s − 1.82e3·23-s + 5.04e3·24-s + 1.40e3·26-s − 4.06e3·27-s + 3.33e3·28-s + 3.41e3·29-s − 7.64e3·31-s + 2.72e3·32-s + 3.24e3·33-s + 1.72e4·34-s + ⋯ |
| L(s) = 1 | + 1.76·2-s + 0.898·3-s + 17/8·4-s + 1.58·6-s + 0.377·7-s + 1.98·8-s − 0.193·9-s + 0.578·11-s + 1.90·12-s + 0.229·13-s + 0.668·14-s + 1.39·16-s + 1.44·17-s − 0.341·18-s − 0.0622·19-s + 0.339·21-s + 1.02·22-s − 0.718·23-s + 1.78·24-s + 0.406·26-s − 1.07·27-s + 0.803·28-s + 0.754·29-s − 1.42·31-s + 0.469·32-s + 0.519·33-s + 2.55·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(7.861741925\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.861741925\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - p^{2} T \) |
| good | 2 | \( 1 - 5 p T + p^{5} T^{2} \) |
| 3 | \( 1 - 14 T + p^{5} T^{2} \) |
| 11 | \( 1 - 232 T + p^{5} T^{2} \) |
| 13 | \( 1 - 140 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1722 T + p^{5} T^{2} \) |
| 19 | \( 1 + 98 T + p^{5} T^{2} \) |
| 23 | \( 1 + 1824 T + p^{5} T^{2} \) |
| 29 | \( 1 - 3418 T + p^{5} T^{2} \) |
| 31 | \( 1 + 7644 T + p^{5} T^{2} \) |
| 37 | \( 1 - 10398 T + p^{5} T^{2} \) |
| 41 | \( 1 + 17962 T + p^{5} T^{2} \) |
| 43 | \( 1 + 10880 T + p^{5} T^{2} \) |
| 47 | \( 1 + 9324 T + p^{5} T^{2} \) |
| 53 | \( 1 + 2262 T + p^{5} T^{2} \) |
| 59 | \( 1 + 2730 T + p^{5} T^{2} \) |
| 61 | \( 1 - 25648 T + p^{5} T^{2} \) |
| 67 | \( 1 - 48404 T + p^{5} T^{2} \) |
| 71 | \( 1 + 58560 T + p^{5} T^{2} \) |
| 73 | \( 1 + 68082 T + p^{5} T^{2} \) |
| 79 | \( 1 - 31784 T + p^{5} T^{2} \) |
| 83 | \( 1 - 20538 T + p^{5} T^{2} \) |
| 89 | \( 1 + 50582 T + p^{5} T^{2} \) |
| 97 | \( 1 - 58506 T + p^{5} T^{2} \) |
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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
−12.00612503520481003014390500628, −11.34705271128151429025700355692, −9.959815623940900851218828111402, −8.539979483506156752982280547873, −7.48808886275778372411031332056, −6.20397807893814053100474613341, −5.19817163096014321019995727184, −3.88553469698264429985104906666, −3.09069771676693723400149567648, −1.77211365157158500631676652077,
1.77211365157158500631676652077, 3.09069771676693723400149567648, 3.88553469698264429985104906666, 5.19817163096014321019995727184, 6.20397807893814053100474613341, 7.48808886275778372411031332056, 8.539979483506156752982280547873, 9.959815623940900851218828111402, 11.34705271128151429025700355692, 12.00612503520481003014390500628
