| L(s) = 1 | + 2·2-s + 4·4-s − 7·5-s − 7·7-s + 8·8-s − 14·10-s − 17·11-s + 12·13-s − 14·14-s + 16·16-s − 38·17-s − 43·19-s − 28·20-s − 34·22-s − 131·23-s − 76·25-s + 24·26-s − 28·28-s − 160·29-s + 45·31-s + 32·32-s − 76·34-s + 49·35-s − 331·37-s − 86·38-s − 56·40-s + 111·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.626·5-s − 0.377·7-s + 0.353·8-s − 0.442·10-s − 0.465·11-s + 0.256·13-s − 0.267·14-s + 1/4·16-s − 0.542·17-s − 0.519·19-s − 0.313·20-s − 0.329·22-s − 1.18·23-s − 0.607·25-s + 0.181·26-s − 0.188·28-s − 1.02·29-s + 0.260·31-s + 0.176·32-s − 0.383·34-s + 0.236·35-s − 1.47·37-s − 0.367·38-s − 0.221·40-s + 0.422·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 5 | \( 1 + 7 T + p^{3} T^{2} \) |
| 11 | \( 1 + 17 T + p^{3} T^{2} \) |
| 13 | \( 1 - 12 T + p^{3} T^{2} \) |
| 17 | \( 1 + 38 T + p^{3} T^{2} \) |
| 19 | \( 1 + 43 T + p^{3} T^{2} \) |
| 23 | \( 1 + 131 T + p^{3} T^{2} \) |
| 29 | \( 1 + 160 T + p^{3} T^{2} \) |
| 31 | \( 1 - 45 T + p^{3} T^{2} \) |
| 37 | \( 1 + 331 T + p^{3} T^{2} \) |
| 41 | \( 1 - 111 T + p^{3} T^{2} \) |
| 43 | \( 1 - 230 T + p^{3} T^{2} \) |
| 47 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 53 | \( 1 + 396 T + p^{3} T^{2} \) |
| 59 | \( 1 + 214 T + p^{3} T^{2} \) |
| 61 | \( 1 - 768 T + p^{3} T^{2} \) |
| 67 | \( 1 - 388 T + p^{3} T^{2} \) |
| 71 | \( 1 + 551 T + p^{3} T^{2} \) |
| 73 | \( 1 - 274 T + p^{3} T^{2} \) |
| 79 | \( 1 - 390 T + p^{3} T^{2} \) |
| 83 | \( 1 + 440 T + p^{3} T^{2} \) |
| 89 | \( 1 + 105 T + p^{3} T^{2} \) |
| 97 | \( 1 - 304 T + p^{3} T^{2} \) |
| 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
−10.70333363817675177931319688289, −9.691943594082112265838131776750, −8.447193006420253300816536773743, −7.57860600489704063044246266631, −6.53854650826415694786622362939, −5.57323611747284592479121334112, −4.33334355544235076741880926114, −3.47778021263476618635090420264, −2.08218217058645274011285682155, 0,
2.08218217058645274011285682155, 3.47778021263476618635090420264, 4.33334355544235076741880926114, 5.57323611747284592479121334112, 6.53854650826415694786622362939, 7.57860600489704063044246266631, 8.447193006420253300816536773743, 9.691943594082112265838131776750, 10.70333363817675177931319688289
