| L(s) = 1 | − 1.23·3-s − 2·5-s − 4.62·7-s − 1.46·9-s − 2.14·11-s + 2.47·15-s + 0.464·17-s + 4.62·19-s + 5.73·21-s + 5.53·23-s − 25-s + 5.53·27-s + 3·29-s + 9.25·31-s + 2.66·33-s + 9.25·35-s − 7.73·37-s − 7.19·41-s + 3.71·43-s + 2.92·45-s + 11.7·47-s + 14.3·49-s − 0.575·51-s + 2.53·53-s + 4.29·55-s − 5.73·57-s − 9.58·59-s + ⋯ |
| L(s) = 1 | − 0.715·3-s − 0.894·5-s − 1.74·7-s − 0.488·9-s − 0.647·11-s + 0.639·15-s + 0.112·17-s + 1.06·19-s + 1.25·21-s + 1.15·23-s − 0.200·25-s + 1.06·27-s + 0.557·29-s + 1.66·31-s + 0.463·33-s + 1.56·35-s − 1.27·37-s − 1.12·41-s + 0.566·43-s + 0.436·45-s + 1.71·47-s + 2.05·49-s − 0.0805·51-s + 0.348·53-s + 0.578·55-s − 0.759·57-s − 1.24·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 1.23T + 3T^{2} \) |
| 5 | \( 1 + 2T + 5T^{2} \) |
| 7 | \( 1 + 4.62T + 7T^{2} \) |
| 11 | \( 1 + 2.14T + 11T^{2} \) |
| 17 | \( 1 - 0.464T + 17T^{2} \) |
| 19 | \( 1 - 4.62T + 19T^{2} \) |
| 23 | \( 1 - 5.53T + 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 - 9.25T + 31T^{2} \) |
| 37 | \( 1 + 7.73T + 37T^{2} \) |
| 41 | \( 1 + 7.19T + 41T^{2} \) |
| 43 | \( 1 - 3.71T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 - 2.53T + 53T^{2} \) |
| 59 | \( 1 + 9.58T + 59T^{2} \) |
| 61 | \( 1 - 3T + 61T^{2} \) |
| 67 | \( 1 - 4.62T + 67T^{2} \) |
| 71 | \( 1 + 6.43T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 4.29T + 79T^{2} \) |
| 83 | \( 1 + 4.29T + 83T^{2} \) |
| 89 | \( 1 - 1.19T + 89T^{2} \) |
| 97 | \( 1 + 13.7T + 97T^{2} \) |
| 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
−7.67028055175067437080811384122, −7.01473896863702454587700602065, −6.41976004844199625543147128010, −5.66485689895821009192110380172, −5.03987297724527830968347051936, −4.05030222647526952038492905098, −3.11352139298563440182612829165, −2.80913507372797213764629690077, −0.866009157093505434128931064751, 0,
0.866009157093505434128931064751, 2.80913507372797213764629690077, 3.11352139298563440182612829165, 4.05030222647526952038492905098, 5.03987297724527830968347051936, 5.66485689895821009192110380172, 6.41976004844199625543147128010, 7.01473896863702454587700602065, 7.67028055175067437080811384122
