| L(s) = 1 | + 2.08·2-s + 2.35·4-s + 1.35·5-s − 0.0861·7-s + 0.734·8-s + 2.82·10-s + 11-s + 0.648·13-s − 0.179·14-s − 3.17·16-s − 3.43·17-s + 4.82·19-s + 3.17·20-s + 2.08·22-s − 5.82·23-s − 3.17·25-s + 1.35·26-s − 0.202·28-s − 4.79·29-s + 0.648·31-s − 8.08·32-s − 7.17·34-s − 0.116·35-s + 11.4·37-s + 10.0·38-s + 0.992·40-s − 8.96·41-s + ⋯ |
| L(s) = 1 | + 1.47·2-s + 1.17·4-s + 0.604·5-s − 0.0325·7-s + 0.259·8-s + 0.891·10-s + 0.301·11-s + 0.179·13-s − 0.0480·14-s − 0.793·16-s − 0.833·17-s + 1.10·19-s + 0.710·20-s + 0.444·22-s − 1.21·23-s − 0.634·25-s + 0.265·26-s − 0.0382·28-s − 0.889·29-s + 0.116·31-s − 1.42·32-s − 1.23·34-s − 0.0196·35-s + 1.87·37-s + 1.63·38-s + 0.156·40-s − 1.39·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.781429063\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.781429063\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 2.08T + 2T^{2} \) |
| 5 | \( 1 - 1.35T + 5T^{2} \) |
| 7 | \( 1 + 0.0861T + 7T^{2} \) |
| 13 | \( 1 - 0.648T + 13T^{2} \) |
| 17 | \( 1 + 3.43T + 17T^{2} \) |
| 19 | \( 1 - 4.82T + 19T^{2} \) |
| 23 | \( 1 + 5.82T + 23T^{2} \) |
| 29 | \( 1 + 4.79T + 29T^{2} \) |
| 31 | \( 1 - 0.648T + 31T^{2} \) |
| 37 | \( 1 - 11.4T + 37T^{2} \) |
| 41 | \( 1 + 8.96T + 41T^{2} \) |
| 43 | \( 1 - 9.61T + 43T^{2} \) |
| 47 | \( 1 + 3.87T + 47T^{2} \) |
| 53 | \( 1 - 7.46T + 53T^{2} \) |
| 59 | \( 1 - 6.69T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 + 4.99T + 67T^{2} \) |
| 71 | \( 1 - 7.46T + 71T^{2} \) |
| 73 | \( 1 + 4.87T + 73T^{2} \) |
| 79 | \( 1 - 10.8T + 79T^{2} \) |
| 83 | \( 1 + 1.94T + 83T^{2} \) |
| 89 | \( 1 - 5.04T + 89T^{2} \) |
| 97 | \( 1 - 15.5T + 97T^{2} \) |
| 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
−11.87680262837227645004666660512, −11.26871107343225911511326452847, −9.936869061616895286252604280654, −9.066285800537894460302800818105, −7.62382217260685226722198614097, −6.36239152620566002868748085466, −5.73150203546226272514122088258, −4.59243983255812478315725439385, −3.54166652051635310209101668181, −2.16189025609765922825554669104,
2.16189025609765922825554669104, 3.54166652051635310209101668181, 4.59243983255812478315725439385, 5.73150203546226272514122088258, 6.36239152620566002868748085466, 7.62382217260685226722198614097, 9.066285800537894460302800818105, 9.936869061616895286252604280654, 11.26871107343225911511326452847, 11.87680262837227645004666660512
