| L(s) = 1 | − 4·2-s + 8·4-s − 5·5-s + 18·7-s + 20·10-s − 10·11-s − 13·13-s − 72·14-s − 64·16-s − 46·17-s − 14·19-s − 40·20-s + 40·22-s + 36·23-s + 25·25-s + 52·26-s + 144·28-s + 22·29-s + 42·31-s + 256·32-s + 184·34-s − 90·35-s − 46·37-s + 56·38-s + 226·41-s − 224·43-s − 80·44-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 0.447·5-s + 0.971·7-s + 0.632·10-s − 0.274·11-s − 0.277·13-s − 1.37·14-s − 16-s − 0.656·17-s − 0.169·19-s − 0.447·20-s + 0.387·22-s + 0.326·23-s + 1/5·25-s + 0.392·26-s + 0.971·28-s + 0.140·29-s + 0.243·31-s + 1.41·32-s + 0.928·34-s − 0.434·35-s − 0.204·37-s + 0.239·38-s + 0.860·41-s − 0.794·43-s − 0.274·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 + p T \) |
| 13 | \( 1 + p T \) |
| good | 2 | \( 1 + p^{2} T + p^{3} T^{2} \) |
| 7 | \( 1 - 18 T + p^{3} T^{2} \) |
| 11 | \( 1 + 10 T + p^{3} T^{2} \) |
| 17 | \( 1 + 46 T + p^{3} T^{2} \) |
| 19 | \( 1 + 14 T + p^{3} T^{2} \) |
| 23 | \( 1 - 36 T + p^{3} T^{2} \) |
| 29 | \( 1 - 22 T + p^{3} T^{2} \) |
| 31 | \( 1 - 42 T + p^{3} T^{2} \) |
| 37 | \( 1 + 46 T + p^{3} T^{2} \) |
| 41 | \( 1 - 226 T + p^{3} T^{2} \) |
| 43 | \( 1 + 224 T + p^{3} T^{2} \) |
| 47 | \( 1 - 50 T + p^{3} T^{2} \) |
| 53 | \( 1 - 290 T + p^{3} T^{2} \) |
| 59 | \( 1 + 130 T + p^{3} T^{2} \) |
| 61 | \( 1 - 70 T + p^{3} T^{2} \) |
| 67 | \( 1 + 138 T + p^{3} T^{2} \) |
| 71 | \( 1 - 586 T + p^{3} T^{2} \) |
| 73 | \( 1 + 758 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1068 T + p^{3} T^{2} \) |
| 83 | \( 1 + 378 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1374 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1822 T + p^{3} T^{2} \) |
| 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
−9.773549091070061065259280648306, −8.849601011386170830030077803641, −8.210407677498587265541606989887, −7.52526796910728818858580331562, −6.65628661479450102836246496960, −5.14378964413893869296258157677, −4.21464535763996814880586199641, −2.49407561259259491948805606566, −1.30680614855976483287254705280, 0,
1.30680614855976483287254705280, 2.49407561259259491948805606566, 4.21464535763996814880586199641, 5.14378964413893869296258157677, 6.65628661479450102836246496960, 7.52526796910728818858580331562, 8.210407677498587265541606989887, 8.849601011386170830030077803641, 9.773549091070061065259280648306
