| L(s) = 1 | − 2.52·2-s − 0.648·3-s + 4.35·4-s + 2.39·5-s + 1.63·6-s − 0.583·7-s − 5.94·8-s − 2.57·9-s − 6.04·10-s − 11-s − 2.82·12-s + 6.74·13-s + 1.47·14-s − 1.55·15-s + 6.27·16-s − 4.94·17-s + 6.50·18-s + 2.88·19-s + 10.4·20-s + 0.378·21-s + 2.52·22-s − 3.86·23-s + 3.85·24-s + 0.752·25-s − 17.0·26-s + 3.61·27-s − 2.54·28-s + ⋯ |
| L(s) = 1 | − 1.78·2-s − 0.374·3-s + 2.17·4-s + 1.07·5-s + 0.667·6-s − 0.220·7-s − 2.10·8-s − 0.859·9-s − 1.91·10-s − 0.301·11-s − 0.815·12-s + 1.87·13-s + 0.393·14-s − 0.401·15-s + 1.56·16-s − 1.19·17-s + 1.53·18-s + 0.662·19-s + 2.33·20-s + 0.0825·21-s + 0.537·22-s − 0.806·23-s + 0.786·24-s + 0.150·25-s − 3.33·26-s + 0.696·27-s − 0.480·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + T \) |
| 131 | \( 1 + T \) |
| good | 2 | \( 1 + 2.52T + 2T^{2} \) |
| 3 | \( 1 + 0.648T + 3T^{2} \) |
| 5 | \( 1 - 2.39T + 5T^{2} \) |
| 7 | \( 1 + 0.583T + 7T^{2} \) |
| 13 | \( 1 - 6.74T + 13T^{2} \) |
| 17 | \( 1 + 4.94T + 17T^{2} \) |
| 19 | \( 1 - 2.88T + 19T^{2} \) |
| 23 | \( 1 + 3.86T + 23T^{2} \) |
| 29 | \( 1 + 7.59T + 29T^{2} \) |
| 31 | \( 1 + 8.60T + 31T^{2} \) |
| 37 | \( 1 + 9.90T + 37T^{2} \) |
| 41 | \( 1 - 8.01T + 41T^{2} \) |
| 43 | \( 1 - 7.33T + 43T^{2} \) |
| 47 | \( 1 + 6.45T + 47T^{2} \) |
| 53 | \( 1 - 3.90T + 53T^{2} \) |
| 59 | \( 1 - 8.38T + 59T^{2} \) |
| 61 | \( 1 + 1.94T + 61T^{2} \) |
| 67 | \( 1 + 2.74T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 - 11.8T + 73T^{2} \) |
| 79 | \( 1 - 3.11T + 79T^{2} \) |
| 83 | \( 1 + 9.83T + 83T^{2} \) |
| 89 | \( 1 + 3.73T + 89T^{2} \) |
| 97 | \( 1 - 17.1T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.065426985787844270576952450352, −8.662589223350561249476347822046, −7.72885173412294072951934434657, −6.74798375302988041997256586746, −6.01162392790687030377503046467, −5.53079132167235516811368791800, −3.63504694039144861314715687432, −2.34286862292796827569489496217, −1.49533068407115263470858076652, 0,
1.49533068407115263470858076652, 2.34286862292796827569489496217, 3.63504694039144861314715687432, 5.53079132167235516811368791800, 6.01162392790687030377503046467, 6.74798375302988041997256586746, 7.72885173412294072951934434657, 8.662589223350561249476347822046, 9.065426985787844270576952450352
