| L(s) = 1 | + 3-s + 0.585·5-s + 1.41·7-s + 9-s + 0.828·11-s + 4.82·13-s + 0.585·15-s − 0.828·17-s − 2.82·19-s + 1.41·21-s + 6.82·23-s − 4.65·25-s + 27-s + 4.58·29-s − 7.07·31-s + 0.828·33-s + 0.828·35-s + 0.343·37-s + 4.82·39-s + 6.48·41-s + 1.17·43-s + 0.585·45-s − 4.48·47-s − 5·49-s − 0.828·51-s + 10.2·53-s + 0.485·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.261·5-s + 0.534·7-s + 0.333·9-s + 0.249·11-s + 1.33·13-s + 0.151·15-s − 0.200·17-s − 0.648·19-s + 0.308·21-s + 1.42·23-s − 0.931·25-s + 0.192·27-s + 0.851·29-s − 1.27·31-s + 0.144·33-s + 0.140·35-s + 0.0564·37-s + 0.773·39-s + 1.01·41-s + 0.178·43-s + 0.0873·45-s − 0.654·47-s − 0.714·49-s − 0.116·51-s + 1.40·53-s + 0.0654·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.524724816\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.524724816\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| good | 5 | \( 1 - 0.585T + 5T^{2} \) |
| 7 | \( 1 - 1.41T + 7T^{2} \) |
| 11 | \( 1 - 0.828T + 11T^{2} \) |
| 13 | \( 1 - 4.82T + 13T^{2} \) |
| 17 | \( 1 + 0.828T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 - 6.82T + 23T^{2} \) |
| 29 | \( 1 - 4.58T + 29T^{2} \) |
| 31 | \( 1 + 7.07T + 31T^{2} \) |
| 37 | \( 1 - 0.343T + 37T^{2} \) |
| 41 | \( 1 - 6.48T + 41T^{2} \) |
| 43 | \( 1 - 1.17T + 43T^{2} \) |
| 47 | \( 1 + 4.48T + 47T^{2} \) |
| 53 | \( 1 - 10.2T + 53T^{2} \) |
| 59 | \( 1 + 9.65T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 - 5.65T + 67T^{2} \) |
| 71 | \( 1 - 8.48T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 + 3.17T + 83T^{2} \) |
| 89 | \( 1 + 17.3T + 89T^{2} \) |
| 97 | \( 1 - 3.65T + 97T^{2} \) |
| 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
−9.263247525888969910434106896920, −8.680088304946863075078983045293, −8.034177794536019404695437551618, −7.06217321280785590516066329506, −6.26083503226687910235809933465, −5.32388382876011597818427962489, −4.27645982070683424710230626549, −3.47215109372313747215187451736, −2.28089429096443038666811498393, −1.22361178559583339778249123595,
1.22361178559583339778249123595, 2.28089429096443038666811498393, 3.47215109372313747215187451736, 4.27645982070683424710230626549, 5.32388382876011597818427962489, 6.26083503226687910235809933465, 7.06217321280785590516066329506, 8.034177794536019404695437551618, 8.680088304946863075078983045293, 9.263247525888969910434106896920
