| L(s) = 1 | + 10·2-s − 14·3-s + 68·4-s − 56·5-s − 140·6-s + 49·7-s + 360·8-s − 47·9-s − 560·10-s − 952·12-s + 140·13-s + 490·14-s + 784·15-s + 1.42e3·16-s + 1.72e3·17-s − 470·18-s + 98·19-s − 3.80e3·20-s − 686·21-s + 1.82e3·23-s − 5.04e3·24-s + 11·25-s + 1.40e3·26-s + 4.06e3·27-s + 3.33e3·28-s − 3.41e3·29-s + 7.84e3·30-s + ⋯ |
| L(s) = 1 | + 1.76·2-s − 0.898·3-s + 17/8·4-s − 1.00·5-s − 1.58·6-s + 0.377·7-s + 1.98·8-s − 0.193·9-s − 1.77·10-s − 1.90·12-s + 0.229·13-s + 0.668·14-s + 0.899·15-s + 1.39·16-s + 1.44·17-s − 0.341·18-s + 0.0622·19-s − 2.12·20-s − 0.339·21-s + 0.718·23-s − 1.78·24-s + 0.00351·25-s + 0.406·26-s + 1.07·27-s + 0.803·28-s − 0.754·29-s + 1.59·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 - p^{2} T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 5 p T + p^{5} T^{2} \) |
| 3 | \( 1 + 14 T + p^{5} T^{2} \) |
| 5 | \( 1 + 56 T + p^{5} T^{2} \) |
| 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
−8.913254114677899756762860883946, −7.65880758368829462260931476763, −7.13105292311223700206681695002, −5.89772248890266473213103942456, −5.49713161726879651952384407058, −4.60816359921528401879414805698, −3.72770282092971980280382275333, −2.97205832026213013485672403234, −1.44375030367695146125445990996, 0,
1.44375030367695146125445990996, 2.97205832026213013485672403234, 3.72770282092971980280382275333, 4.60816359921528401879414805698, 5.49713161726879651952384407058, 5.89772248890266473213103942456, 7.13105292311223700206681695002, 7.65880758368829462260931476763, 8.913254114677899756762860883946
