| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s − 5·5-s + 6·6-s − 8·8-s + 9·9-s + 10·10-s − 4·11-s − 12·12-s + 62·13-s + 15·15-s + 16·16-s − 70·17-s − 18·18-s + 6·19-s − 20·20-s + 8·22-s − 70·23-s + 24·24-s + 25·25-s − 124·26-s − 27·27-s − 162·29-s − 30·30-s − 62·31-s − 32·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.109·11-s − 0.288·12-s + 1.32·13-s + 0.258·15-s + 1/4·16-s − 0.998·17-s − 0.235·18-s + 0.0724·19-s − 0.223·20-s + 0.0775·22-s − 0.634·23-s + 0.204·24-s + 1/5·25-s − 0.935·26-s − 0.192·27-s − 1.03·29-s − 0.182·30-s − 0.359·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 + p T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4 T + p^{3} T^{2} \) |
| 13 | \( 1 - 62 T + p^{3} T^{2} \) |
| 17 | \( 1 + 70 T + p^{3} T^{2} \) |
| 19 | \( 1 - 6 T + p^{3} T^{2} \) |
| 23 | \( 1 + 70 T + p^{3} T^{2} \) |
| 29 | \( 1 + 162 T + p^{3} T^{2} \) |
| 31 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 37 | \( 1 - 218 T + p^{3} T^{2} \) |
| 41 | \( 1 - 130 T + p^{3} T^{2} \) |
| 43 | \( 1 - 232 T + p^{3} T^{2} \) |
| 47 | \( 1 - 304 T + p^{3} T^{2} \) |
| 53 | \( 1 + 380 T + p^{3} T^{2} \) |
| 59 | \( 1 - 376 T + p^{3} T^{2} \) |
| 61 | \( 1 - 56 T + p^{3} T^{2} \) |
| 67 | \( 1 + 952 T + p^{3} T^{2} \) |
| 71 | \( 1 - 708 T + p^{3} T^{2} \) |
| 73 | \( 1 - 682 T + p^{3} T^{2} \) |
| 79 | \( 1 + 8 p T + p^{3} T^{2} \) |
| 83 | \( 1 - 244 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1198 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1206 T + p^{3} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.784736602372790476999757894338, −7.949856339469292556343884030319, −7.22113135103577952405066473776, −6.29612550000783683805327664247, −5.69896700888903782718509006385, −4.42381615319785833742108794566, −3.61206049189477620300415611504, −2.25332418615704155282446712927, −1.08069409717834126094361132377, 0,
1.08069409717834126094361132377, 2.25332418615704155282446712927, 3.61206049189477620300415611504, 4.42381615319785833742108794566, 5.69896700888903782718509006385, 6.29612550000783683805327664247, 7.22113135103577952405066473776, 7.949856339469292556343884030319, 8.784736602372790476999757894338
