| L(s) = 1 | − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s − 5·11-s − 13-s − 14-s + 16-s + 3·17-s − 19-s + 20-s + 5·22-s − 3·23-s − 4·25-s + 26-s + 28-s − 9·29-s + 4·31-s − 32-s − 3·34-s + 35-s − 11·37-s + 38-s − 40-s − 5·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 0.316·10-s − 1.50·11-s − 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s − 0.229·19-s + 0.223·20-s + 1.06·22-s − 0.625·23-s − 4/5·25-s + 0.196·26-s + 0.188·28-s − 1.67·29-s + 0.718·31-s − 0.176·32-s − 0.514·34-s + 0.169·35-s − 1.80·37-s + 0.162·38-s − 0.158·40-s − 0.762·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
| 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
−9.067467943273849313007184265002, −7.935932481491448532694078819912, −7.80015052507622603521937526703, −6.69232029580076888718309416404, −5.63709318901757604410923700064, −5.14572588019451526117366164450, −3.74734603047492382743471605599, −2.56045395592179615052900667030, −1.71010590596796076771996481178, 0,
1.71010590596796076771996481178, 2.56045395592179615052900667030, 3.74734603047492382743471605599, 5.14572588019451526117366164450, 5.63709318901757604410923700064, 6.69232029580076888718309416404, 7.80015052507622603521937526703, 7.935932481491448532694078819912, 9.067467943273849313007184265002
