| L(s) = 1 | − 4·2-s + 11·3-s + 16·4-s − 44·6-s − 49·7-s − 64·8-s − 122·9-s − 267·11-s + 176·12-s + 1.08e3·13-s + 196·14-s + 256·16-s + 513·17-s + 488·18-s − 802·19-s − 539·21-s + 1.06e3·22-s + 1.29e3·23-s − 704·24-s − 4.34e3·26-s − 4.01e3·27-s − 784·28-s + 1.77e3·29-s − 2.58e3·31-s − 1.02e3·32-s − 2.93e3·33-s − 2.05e3·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.705·3-s + 1/2·4-s − 0.498·6-s − 0.377·7-s − 0.353·8-s − 0.502·9-s − 0.665·11-s + 0.352·12-s + 1.78·13-s + 0.267·14-s + 1/4·16-s + 0.430·17-s + 0.355·18-s − 0.509·19-s − 0.266·21-s + 0.470·22-s + 0.508·23-s − 0.249·24-s − 1.26·26-s − 1.05·27-s − 0.188·28-s + 0.392·29-s − 0.482·31-s − 0.176·32-s − 0.469·33-s − 0.304·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{2} T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + p^{2} T \) |
| good | 3 | \( 1 - 11 T + p^{5} T^{2} \) |
| 11 | \( 1 + 267 T + p^{5} T^{2} \) |
| 13 | \( 1 - 1087 T + p^{5} T^{2} \) |
| 17 | \( 1 - 513 T + p^{5} T^{2} \) |
| 19 | \( 1 + 802 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1290 T + p^{5} T^{2} \) |
| 29 | \( 1 - 1779 T + p^{5} T^{2} \) |
| 31 | \( 1 + 2584 T + p^{5} T^{2} \) |
| 37 | \( 1 + 13862 T + p^{5} T^{2} \) |
| 41 | \( 1 + 11904 T + p^{5} T^{2} \) |
| 43 | \( 1 - 598 T + p^{5} T^{2} \) |
| 47 | \( 1 - 17019 T + p^{5} T^{2} \) |
| 53 | \( 1 + 27852 T + p^{5} T^{2} \) |
| 59 | \( 1 - 30912 T + p^{5} T^{2} \) |
| 61 | \( 1 + 1780 T + p^{5} T^{2} \) |
| 67 | \( 1 + 25052 T + p^{5} T^{2} \) |
| 71 | \( 1 + 51984 T + p^{5} T^{2} \) |
| 73 | \( 1 + 47690 T + p^{5} T^{2} \) |
| 79 | \( 1 + 102121 T + p^{5} T^{2} \) |
| 83 | \( 1 - 83676 T + p^{5} T^{2} \) |
| 89 | \( 1 + 32400 T + p^{5} T^{2} \) |
| 97 | \( 1 - 148645 T + p^{5} T^{2} \) |
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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
−10.20242003240162580897755457699, −8.909594847363771695660584004193, −8.607945881738578257862504015549, −7.61416245657015220848157960762, −6.45964758154176663310586217606, −5.48348114496744528615004206462, −3.69565242178110963316303699540, −2.83837887333414565772330317588, −1.49489331137436378747435832505, 0,
1.49489331137436378747435832505, 2.83837887333414565772330317588, 3.69565242178110963316303699540, 5.48348114496744528615004206462, 6.45964758154176663310586217606, 7.61416245657015220848157960762, 8.607945881738578257862504015549, 8.909594847363771695660584004193, 10.20242003240162580897755457699
