| L(s) = 1 | − 1.73·2-s + 3-s + 0.999·4-s − 1.73·6-s + 7-s + 1.73·8-s + 9-s − 4·11-s + 0.999·12-s − 2.26·13-s − 1.73·14-s − 5·16-s + 7.19·17-s − 1.73·18-s − 1.73·19-s + 21-s + 6.92·22-s − 2.46·23-s + 1.73·24-s + 3.92·26-s + 27-s + 0.999·28-s − 6.46·29-s + 3.46·31-s + 5.19·32-s − 4·33-s − 12.4·34-s + ⋯ |
| L(s) = 1 | − 1.22·2-s + 0.577·3-s + 0.499·4-s − 0.707·6-s + 0.377·7-s + 0.612·8-s + 0.333·9-s − 1.20·11-s + 0.288·12-s − 0.629·13-s − 0.462·14-s − 1.25·16-s + 1.74·17-s − 0.408·18-s − 0.397·19-s + 0.218·21-s + 1.47·22-s − 0.513·23-s + 0.353·24-s + 0.770·26-s + 0.192·27-s + 0.188·28-s − 1.20·29-s + 0.622·31-s + 0.918·32-s − 0.696·33-s − 2.13·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 2.26T + 13T^{2} \) |
| 17 | \( 1 - 7.19T + 17T^{2} \) |
| 19 | \( 1 + 1.73T + 19T^{2} \) |
| 23 | \( 1 + 2.46T + 23T^{2} \) |
| 29 | \( 1 + 6.46T + 29T^{2} \) |
| 31 | \( 1 - 3.46T + 31T^{2} \) |
| 37 | \( 1 + 4.53T + 37T^{2} \) |
| 41 | \( 1 - 0.464T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 53 | \( 1 + 14.1T + 53T^{2} \) |
| 59 | \( 1 + 4.80T + 59T^{2} \) |
| 61 | \( 1 - 7.92T + 61T^{2} \) |
| 67 | \( 1 + 4.92T + 67T^{2} \) |
| 71 | \( 1 + 7.19T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + 1.46T + 79T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 + 1.46T + 97T^{2} \) |
| 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
−8.217198380165869182747558166138, −7.52263222634567871047367943775, −7.43751662001673011341206139141, −6.01445117065334862304234225611, −5.14591045499074417122822406934, −4.36170090772263821306866599562, −3.22068974932638116197910080289, −2.27957819533470281315089801318, −1.35971850042607192829234392338, 0,
1.35971850042607192829234392338, 2.27957819533470281315089801318, 3.22068974932638116197910080289, 4.36170090772263821306866599562, 5.14591045499074417122822406934, 6.01445117065334862304234225611, 7.43751662001673011341206139141, 7.52263222634567871047367943775, 8.217198380165869182747558166138
