| L(s) = 1 | + 4·2-s + 8·3-s + 16·4-s + 10·5-s + 32·6-s − 49·7-s + 64·8-s − 179·9-s + 40·10-s − 340·11-s + 128·12-s − 294·13-s − 196·14-s + 80·15-s + 256·16-s + 1.22e3·17-s − 716·18-s + 2.43e3·19-s + 160·20-s − 392·21-s − 1.36e3·22-s + 2.00e3·23-s + 512·24-s − 3.02e3·25-s − 1.17e3·26-s − 3.37e3·27-s − 784·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.513·3-s + 1/2·4-s + 0.178·5-s + 0.362·6-s − 0.377·7-s + 0.353·8-s − 0.736·9-s + 0.126·10-s − 0.847·11-s + 0.256·12-s − 0.482·13-s − 0.267·14-s + 0.0918·15-s + 1/4·16-s + 1.02·17-s − 0.520·18-s + 1.54·19-s + 0.0894·20-s − 0.193·21-s − 0.599·22-s + 0.788·23-s + 0.181·24-s − 0.967·25-s − 0.341·26-s − 0.891·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.965737277\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.965737277\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 7 | \( 1 + p^{2} T \) |
| good | 3 | \( 1 - 8 T + p^{5} T^{2} \) |
| 5 | \( 1 - 2 p T + p^{5} T^{2} \) |
| 11 | \( 1 + 340 T + p^{5} T^{2} \) |
| 13 | \( 1 + 294 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1226 T + p^{5} T^{2} \) |
| 19 | \( 1 - 128 p T + p^{5} T^{2} \) |
| 23 | \( 1 - 2000 T + p^{5} T^{2} \) |
| 29 | \( 1 + 6746 T + p^{5} T^{2} \) |
| 31 | \( 1 - 8856 T + p^{5} T^{2} \) |
| 37 | \( 1 - 9182 T + p^{5} T^{2} \) |
| 41 | \( 1 + 14574 T + p^{5} T^{2} \) |
| 43 | \( 1 - 8108 T + p^{5} T^{2} \) |
| 47 | \( 1 + 312 T + p^{5} T^{2} \) |
| 53 | \( 1 + 14634 T + p^{5} T^{2} \) |
| 59 | \( 1 + 27656 T + p^{5} T^{2} \) |
| 61 | \( 1 - 34338 T + p^{5} T^{2} \) |
| 67 | \( 1 - 12316 T + p^{5} T^{2} \) |
| 71 | \( 1 - 520 p T + p^{5} T^{2} \) |
| 73 | \( 1 + 61718 T + p^{5} T^{2} \) |
| 79 | \( 1 + 64752 T + p^{5} T^{2} \) |
| 83 | \( 1 + 77056 T + p^{5} T^{2} \) |
| 89 | \( 1 + 8166 T + p^{5} T^{2} \) |
| 97 | \( 1 - 20650 T + p^{5} T^{2} \) |
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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
−18.82013620948375146079219521011, −17.04989314969958396275275151463, −15.60924522120988394847505812461, −14.31295896588001224903037778226, −13.22546256367041393676775777725, −11.66296854692920815464762279104, −9.767751901650786651119530298980, −7.71778567378258372481301141794, −5.53143925643359300485509392931, −3.00868916649922978616976932493,
3.00868916649922978616976932493, 5.53143925643359300485509392931, 7.71778567378258372481301141794, 9.767751901650786651119530298980, 11.66296854692920815464762279104, 13.22546256367041393676775777725, 14.31295896588001224903037778226, 15.60924522120988394847505812461, 17.04989314969958396275275151463, 18.82013620948375146079219521011
