| L(s) = 1 | + 2-s + 3·3-s + 4-s + 3·6-s + 7-s + 8-s + 6·9-s + 3·12-s − 2·13-s + 14-s + 16-s + 7·17-s + 6·18-s + 4·19-s + 3·21-s − 4·23-s + 3·24-s − 2·26-s + 9·27-s + 28-s + 2·29-s + 31-s + 32-s + 7·34-s + 6·36-s − 2·37-s + 4·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.73·3-s + 1/2·4-s + 1.22·6-s + 0.377·7-s + 0.353·8-s + 2·9-s + 0.866·12-s − 0.554·13-s + 0.267·14-s + 1/4·16-s + 1.69·17-s + 1.41·18-s + 0.917·19-s + 0.654·21-s − 0.834·23-s + 0.612·24-s − 0.392·26-s + 1.73·27-s + 0.188·28-s + 0.371·29-s + 0.179·31-s + 0.176·32-s + 1.20·34-s + 36-s − 0.328·37-s + 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.40128326\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.40128326\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 293 | \( 1 + T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 5 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
| 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
−13.83002348437367, −13.36680083337880, −12.99688945948559, −12.20315065655958, −12.01579463058352, −11.56087722913514, −10.59147812445121, −10.13561767579777, −9.860154041825514, −9.273287018453793, −8.650840338305495, −8.172045398595461, −7.734764809242261, −7.323411173110374, −6.910054200299558, −6.012271335312487, −5.445178020972677, −4.955291100491178, −4.216075993120054, −3.783410803116784, −3.193556417047603, −2.807891138351247, −2.141550618871290, −1.562658630364033, −0.8500919470014782,
0.8500919470014782, 1.562658630364033, 2.141550618871290, 2.807891138351247, 3.193556417047603, 3.783410803116784, 4.216075993120054, 4.955291100491178, 5.445178020972677, 6.012271335312487, 6.910054200299558, 7.323411173110374, 7.734764809242261, 8.172045398595461, 8.650840338305495, 9.273287018453793, 9.860154041825514, 10.13561767579777, 10.59147812445121, 11.56087722913514, 12.01579463058352, 12.20315065655958, 12.99688945948559, 13.36680083337880, 13.83002348437367
