| L(s) = 1 | + 2·2-s − 7·3-s + 4·4-s + 5·5-s − 14·6-s + 8·8-s + 22·9-s + 10·10-s − 33·11-s − 28·12-s + 43·13-s − 35·15-s + 16·16-s − 111·17-s + 44·18-s + 70·19-s + 20·20-s − 66·22-s + 42·23-s − 56·24-s + 25·25-s + 86·26-s + 35·27-s − 225·29-s − 70·30-s + 88·31-s + 32·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.34·3-s + 1/2·4-s + 0.447·5-s − 0.952·6-s + 0.353·8-s + 0.814·9-s + 0.316·10-s − 0.904·11-s − 0.673·12-s + 0.917·13-s − 0.602·15-s + 1/4·16-s − 1.58·17-s + 0.576·18-s + 0.845·19-s + 0.223·20-s − 0.639·22-s + 0.380·23-s − 0.476·24-s + 1/5·25-s + 0.648·26-s + 0.249·27-s − 1.44·29-s − 0.426·30-s + 0.509·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 7 T + p^{3} T^{2} \) |
| 11 | \( 1 + 3 p T + p^{3} T^{2} \) |
| 13 | \( 1 - 43 T + p^{3} T^{2} \) |
| 17 | \( 1 + 111 T + p^{3} T^{2} \) |
| 19 | \( 1 - 70 T + p^{3} T^{2} \) |
| 23 | \( 1 - 42 T + p^{3} T^{2} \) |
| 29 | \( 1 + 225 T + p^{3} T^{2} \) |
| 31 | \( 1 - 88 T + p^{3} T^{2} \) |
| 37 | \( 1 + 34 T + p^{3} T^{2} \) |
| 41 | \( 1 + 432 T + p^{3} T^{2} \) |
| 43 | \( 1 + 178 T + p^{3} T^{2} \) |
| 47 | \( 1 + 411 T + p^{3} T^{2} \) |
| 53 | \( 1 + 708 T + p^{3} T^{2} \) |
| 59 | \( 1 + 480 T + p^{3} T^{2} \) |
| 61 | \( 1 + 812 T + p^{3} T^{2} \) |
| 67 | \( 1 - 596 T + p^{3} T^{2} \) |
| 71 | \( 1 - 432 T + p^{3} T^{2} \) |
| 73 | \( 1 - 358 T + p^{3} T^{2} \) |
| 79 | \( 1 - 425 T + p^{3} T^{2} \) |
| 83 | \( 1 + 972 T + p^{3} T^{2} \) |
| 89 | \( 1 + 960 T + p^{3} T^{2} \) |
| 97 | \( 1 - 709 T + p^{3} T^{2} \) |
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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
−10.54910345808712796702091380829, −9.418027316248037874551009339162, −8.152668305399606981443608572015, −6.86048310687503810340152392540, −6.21900072901172113409713365931, −5.32547497348909143092960019509, −4.68818915848821752543134777441, −3.20875662775073524290471533045, −1.67251415558945617095363690021, 0,
1.67251415558945617095363690021, 3.20875662775073524290471533045, 4.68818915848821752543134777441, 5.32547497348909143092960019509, 6.21900072901172113409713365931, 6.86048310687503810340152392540, 8.152668305399606981443608572015, 9.418027316248037874551009339162, 10.54910345808712796702091380829
