| L(s) = 1 | + 2-s + 0.714·3-s + 4-s − 0.233·5-s + 0.714·6-s + 8-s − 2.48·9-s − 0.233·10-s + 0.293·11-s + 0.714·12-s − 4.15·13-s − 0.166·15-s + 16-s − 0.609·17-s − 2.48·18-s − 6.67·19-s − 0.233·20-s + 0.293·22-s − 1.48·23-s + 0.714·24-s − 4.94·25-s − 4.15·26-s − 3.92·27-s − 29-s − 0.166·30-s + 1.35·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.412·3-s + 0.5·4-s − 0.104·5-s + 0.291·6-s + 0.353·8-s − 0.829·9-s − 0.0737·10-s + 0.0884·11-s + 0.206·12-s − 1.15·13-s − 0.0430·15-s + 0.250·16-s − 0.147·17-s − 0.586·18-s − 1.53·19-s − 0.0521·20-s + 0.0625·22-s − 0.310·23-s + 0.145·24-s − 0.989·25-s − 0.815·26-s − 0.754·27-s − 0.185·29-s − 0.0304·30-s + 0.242·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 0.714T + 3T^{2} \) |
| 5 | \( 1 + 0.233T + 5T^{2} \) |
| 11 | \( 1 - 0.293T + 11T^{2} \) |
| 13 | \( 1 + 4.15T + 13T^{2} \) |
| 17 | \( 1 + 0.609T + 17T^{2} \) |
| 19 | \( 1 + 6.67T + 19T^{2} \) |
| 23 | \( 1 + 1.48T + 23T^{2} \) |
| 31 | \( 1 - 1.35T + 31T^{2} \) |
| 37 | \( 1 - 0.637T + 37T^{2} \) |
| 41 | \( 1 + 3.31T + 41T^{2} \) |
| 43 | \( 1 + 3.29T + 43T^{2} \) |
| 47 | \( 1 + 2.18T + 47T^{2} \) |
| 53 | \( 1 - 11.9T + 53T^{2} \) |
| 59 | \( 1 + 4.30T + 59T^{2} \) |
| 61 | \( 1 + 3.64T + 61T^{2} \) |
| 67 | \( 1 - 4.95T + 67T^{2} \) |
| 71 | \( 1 - 4.21T + 71T^{2} \) |
| 73 | \( 1 + 5.84T + 73T^{2} \) |
| 79 | \( 1 - 2.10T + 79T^{2} \) |
| 83 | \( 1 + 6.73T + 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 + 5.70T + 97T^{2} \) |
| 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
−8.314347335169829874266284637765, −7.68274190144318904770596166034, −6.79508534274820881473627590430, −6.05749251166054018855601490881, −5.26472880568909878577360395396, −4.40341404326942714998195828219, −3.63859373649584479674640626054, −2.62288994853444808375162110032, −1.98700655484632523142578846600, 0,
1.98700655484632523142578846600, 2.62288994853444808375162110032, 3.63859373649584479674640626054, 4.40341404326942714998195828219, 5.26472880568909878577360395396, 6.05749251166054018855601490881, 6.79508534274820881473627590430, 7.68274190144318904770596166034, 8.314347335169829874266284637765
