| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 7-s − 8-s + 9-s − 2·11-s − 12-s + 5·13-s − 14-s + 16-s − 17-s − 18-s + 19-s − 21-s + 2·22-s + 5·23-s + 24-s − 5·26-s − 27-s + 28-s + 9·29-s + 3·31-s − 32-s + 2·33-s + 34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.603·11-s − 0.288·12-s + 1.38·13-s − 0.267·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.229·19-s − 0.218·21-s + 0.426·22-s + 1.04·23-s + 0.204·24-s − 0.980·26-s − 0.192·27-s + 0.188·28-s + 1.67·29-s + 0.538·31-s − 0.176·32-s + 0.348·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
| 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
−15.78570938822357, −15.66120552581310, −15.16750467553315, −14.25806978351640, −13.68833498181171, −13.23623180043995, −12.56744808628541, −11.76689404237077, −11.63355503439997, −10.80529843035086, −10.37160569926445, −10.13713789935495, −8.974354376563488, −8.728634841564540, −8.182389143852626, −7.386025662492733, −6.878336146380022, −6.247222654638402, −5.650627965032380, −4.944124408019179, −4.343180711038741, −3.322880639069419, −2.760388542924201, −1.602235755037530, −1.109574347031903, 0,
1.109574347031903, 1.602235755037530, 2.760388542924201, 3.322880639069419, 4.343180711038741, 4.944124408019179, 5.650627965032380, 6.247222654638402, 6.878336146380022, 7.386025662492733, 8.182389143852626, 8.728634841564540, 8.974354376563488, 10.13713789935495, 10.37160569926445, 10.80529843035086, 11.63355503439997, 11.76689404237077, 12.56744808628541, 13.23623180043995, 13.68833498181171, 14.25806978351640, 15.16750467553315, 15.66120552581310, 15.78570938822357
