| L(s) = 1 | + 4·2-s + 16·4-s + 25·5-s + 49·7-s + 64·8-s + 100·10-s + 154·11-s − 404·13-s + 196·14-s + 256·16-s − 2.18e3·17-s − 2.49e3·19-s + 400·20-s + 616·22-s + 3.47e3·23-s + 625·25-s − 1.61e3·26-s + 784·28-s − 5.95e3·29-s − 6.41e3·31-s + 1.02e3·32-s − 8.72e3·34-s + 1.22e3·35-s − 1.11e4·37-s − 9.97e3·38-s + 1.60e3·40-s − 7.83e3·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s + 0.353·8-s + 0.316·10-s + 0.383·11-s − 0.663·13-s + 0.267·14-s + 1/4·16-s − 1.83·17-s − 1.58·19-s + 0.223·20-s + 0.271·22-s + 1.36·23-s + 1/5·25-s − 0.468·26-s + 0.188·28-s − 1.31·29-s − 1.19·31-s + 0.176·32-s − 1.29·34-s + 0.169·35-s − 1.33·37-s − 1.12·38-s + 0.158·40-s − 0.727·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p^{2} T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - p^{2} T \) |
| 7 | \( 1 - p^{2} T \) |
| good | 11 | \( 1 - 14 p T + p^{5} T^{2} \) |
| 13 | \( 1 + 404 T + p^{5} T^{2} \) |
| 17 | \( 1 + 2182 T + p^{5} T^{2} \) |
| 19 | \( 1 + 2494 T + p^{5} T^{2} \) |
| 23 | \( 1 - 3472 T + p^{5} T^{2} \) |
| 29 | \( 1 + 5958 T + p^{5} T^{2} \) |
| 31 | \( 1 + 6410 T + p^{5} T^{2} \) |
| 37 | \( 1 + 11150 T + p^{5} T^{2} \) |
| 41 | \( 1 + 7834 T + p^{5} T^{2} \) |
| 43 | \( 1 - 16236 T + p^{5} T^{2} \) |
| 47 | \( 1 - 2800 T + p^{5} T^{2} \) |
| 53 | \( 1 - 30924 T + p^{5} T^{2} \) |
| 59 | \( 1 - 11536 T + p^{5} T^{2} \) |
| 61 | \( 1 + 38834 T + p^{5} T^{2} \) |
| 67 | \( 1 + 48756 T + p^{5} T^{2} \) |
| 71 | \( 1 - 77882 T + p^{5} T^{2} \) |
| 73 | \( 1 + 47540 T + p^{5} T^{2} \) |
| 79 | \( 1 + 36480 T + p^{5} T^{2} \) |
| 83 | \( 1 + 25716 T + p^{5} T^{2} \) |
| 89 | \( 1 - 100826 T + p^{5} T^{2} \) |
| 97 | \( 1 + 89024 T + p^{5} T^{2} \) |
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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
−9.213780134245543314051092778328, −8.707836052886545068882791530478, −7.27971077651217620963119605991, −6.69681842170163765493429424063, −5.63836950642480222807778915389, −4.73029107932236324002035760521, −3.89966676292265086628198164618, −2.47839139944538123200680794222, −1.73189293514939997114613565226, 0,
1.73189293514939997114613565226, 2.47839139944538123200680794222, 3.89966676292265086628198164618, 4.73029107932236324002035760521, 5.63836950642480222807778915389, 6.69681842170163765493429424063, 7.27971077651217620963119605991, 8.707836052886545068882791530478, 9.213780134245543314051092778328
