| L(s) = 1 | − 1.73·2-s + 0.999·4-s − 2·7-s + 1.73·8-s − 3.46·11-s + 13-s + 3.46·14-s − 5·16-s − 5.19·17-s + 2·19-s + 5.99·22-s − 3.46·23-s − 1.73·26-s − 1.99·28-s + 1.73·29-s + 8·31-s + 5.19·32-s + 9·34-s + 7·37-s − 3.46·38-s − 6.92·41-s − 2·43-s − 3.46·44-s + 5.99·46-s − 6.92·47-s − 3·49-s + 0.999·52-s + ⋯ |
| L(s) = 1 | − 1.22·2-s + 0.499·4-s − 0.755·7-s + 0.612·8-s − 1.04·11-s + 0.277·13-s + 0.925·14-s − 1.25·16-s − 1.26·17-s + 0.458·19-s + 1.27·22-s − 0.722·23-s − 0.339·26-s − 0.377·28-s + 0.321·29-s + 1.43·31-s + 0.918·32-s + 1.54·34-s + 1.15·37-s − 0.561·38-s − 1.08·41-s − 0.304·43-s − 0.522·44-s + 0.884·46-s − 1.01·47-s − 0.428·49-s + 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5159078433\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5159078433\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 + 5.19T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 - 1.73T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 7T + 37T^{2} \) |
| 41 | \( 1 + 6.92T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 + 7T + 61T^{2} \) |
| 67 | \( 1 - 10T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 - 7T + 73T^{2} \) |
| 79 | \( 1 - 2T + 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 + 5.19T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
| 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
−9.184223208127508094098709396938, −8.310513875915312226017998862339, −7.947618380449350930263817051252, −6.88594086988018500425242975268, −6.32232728731598886210263906668, −5.11771642284714285640371549606, −4.26475609025031521928762946341, −3.01046826617625559488310943354, −2.01476746180984598759487958200, −0.55671524723422210844814596870,
0.55671524723422210844814596870, 2.01476746180984598759487958200, 3.01046826617625559488310943354, 4.26475609025031521928762946341, 5.11771642284714285640371549606, 6.32232728731598886210263906668, 6.88594086988018500425242975268, 7.947618380449350930263817051252, 8.310513875915312226017998862339, 9.184223208127508094098709396938
