| L(s) = 1 | − 4·2-s − 5·3-s + 16·4-s − 25·5-s + 20·6-s − 64·8-s − 218·9-s + 100·10-s + 198·11-s − 80·12-s − 340·13-s + 125·15-s + 256·16-s + 1.84e3·17-s + 872·18-s − 1.21e3·19-s − 400·20-s − 792·22-s + 2.82e3·23-s + 320·24-s + 625·25-s + 1.36e3·26-s + 2.30e3·27-s − 4.53e3·29-s − 500·30-s − 712·31-s − 1.02e3·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.320·3-s + 1/2·4-s − 0.447·5-s + 0.226·6-s − 0.353·8-s − 0.897·9-s + 0.316·10-s + 0.493·11-s − 0.160·12-s − 0.557·13-s + 0.143·15-s + 1/4·16-s + 1.55·17-s + 0.634·18-s − 0.768·19-s − 0.223·20-s − 0.348·22-s + 1.11·23-s + 0.113·24-s + 1/5·25-s + 0.394·26-s + 0.608·27-s − 1.00·29-s − 0.101·30-s − 0.133·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p^{2} T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 5 T + p^{5} T^{2} \) |
| 11 | \( 1 - 18 p T + p^{5} T^{2} \) |
| 13 | \( 1 + 340 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1848 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1210 T + p^{5} T^{2} \) |
| 23 | \( 1 - 2823 T + p^{5} T^{2} \) |
| 29 | \( 1 + 4539 T + p^{5} T^{2} \) |
| 31 | \( 1 + 712 T + p^{5} T^{2} \) |
| 37 | \( 1 + 7324 T + p^{5} T^{2} \) |
| 41 | \( 1 - 15633 T + p^{5} T^{2} \) |
| 43 | \( 1 - 15827 T + p^{5} T^{2} \) |
| 47 | \( 1 + 3192 T + p^{5} T^{2} \) |
| 53 | \( 1 + 20046 T + p^{5} T^{2} \) |
| 59 | \( 1 - 23046 T + p^{5} T^{2} \) |
| 61 | \( 1 + 379 T + p^{5} T^{2} \) |
| 67 | \( 1 + 35473 T + p^{5} T^{2} \) |
| 71 | \( 1 - 71814 T + p^{5} T^{2} \) |
| 73 | \( 1 - 31664 T + p^{5} T^{2} \) |
| 79 | \( 1 - 8534 T + p^{5} T^{2} \) |
| 83 | \( 1 + 106551 T + p^{5} T^{2} \) |
| 89 | \( 1 + 12303 T + p^{5} T^{2} \) |
| 97 | \( 1 + 102802 T + p^{5} T^{2} \) |
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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
−9.615870674164418658863117830815, −8.871562357472780033269785115168, −7.930195139847184114257882536551, −7.14990445350704787974502615024, −6.05986591078844456835430972606, −5.15054978845127285450422846705, −3.70984437855153237747120140118, −2.60339062353864051727882878826, −1.10518105160673696012672215949, 0,
1.10518105160673696012672215949, 2.60339062353864051727882878826, 3.70984437855153237747120140118, 5.15054978845127285450422846705, 6.05986591078844456835430972606, 7.14990445350704787974502615024, 7.930195139847184114257882536551, 8.871562357472780033269785115168, 9.615870674164418658863117830815
