| L(s) = 1 | + 2.53·2-s + 4.41·4-s + 0.879·5-s + 4.41·7-s + 6.10·8-s + 2.22·10-s + 3.71·11-s − 3.12·13-s + 11.1·14-s + 6.63·16-s − 2.04·19-s + 3.87·20-s + 9.41·22-s + 0.162·23-s − 4.22·25-s − 7.90·26-s + 19.4·28-s − 8.33·29-s + 1.77·31-s + 4.59·32-s + 3.87·35-s − 2.73·37-s − 5.17·38-s + 5.36·40-s + 2.58·41-s − 11.7·43-s + 16.3·44-s + ⋯ |
| L(s) = 1 | + 1.79·2-s + 2.20·4-s + 0.393·5-s + 1.66·7-s + 2.15·8-s + 0.704·10-s + 1.12·11-s − 0.865·13-s + 2.98·14-s + 1.65·16-s − 0.468·19-s + 0.867·20-s + 2.00·22-s + 0.0338·23-s − 0.845·25-s − 1.54·26-s + 3.67·28-s − 1.54·29-s + 0.318·31-s + 0.812·32-s + 0.655·35-s − 0.449·37-s − 0.838·38-s + 0.849·40-s + 0.404·41-s − 1.79·43-s + 2.47·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.061968286\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.061968286\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 2.53T + 2T^{2} \) |
| 5 | \( 1 - 0.879T + 5T^{2} \) |
| 7 | \( 1 - 4.41T + 7T^{2} \) |
| 11 | \( 1 - 3.71T + 11T^{2} \) |
| 13 | \( 1 + 3.12T + 13T^{2} \) |
| 19 | \( 1 + 2.04T + 19T^{2} \) |
| 23 | \( 1 - 0.162T + 23T^{2} \) |
| 29 | \( 1 + 8.33T + 29T^{2} \) |
| 31 | \( 1 - 1.77T + 31T^{2} \) |
| 37 | \( 1 + 2.73T + 37T^{2} \) |
| 41 | \( 1 - 2.58T + 41T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 + 3.26T + 47T^{2} \) |
| 53 | \( 1 - 6.46T + 53T^{2} \) |
| 59 | \( 1 + 7.23T + 59T^{2} \) |
| 61 | \( 1 - 2.50T + 61T^{2} \) |
| 67 | \( 1 + 4.61T + 67T^{2} \) |
| 71 | \( 1 - 1.12T + 71T^{2} \) |
| 73 | \( 1 + 2.77T + 73T^{2} \) |
| 79 | \( 1 + 3.45T + 79T^{2} \) |
| 83 | \( 1 - 14.8T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 - 4.30T + 97T^{2} \) |
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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
−8.776827255747300284110583645747, −7.80508399651374709383839404674, −7.14771969117923871254393122366, −6.30562534794010891274807107274, −5.53655916663109232849192235123, −4.88278545626254278974863093031, −4.27595667273238433856864492537, −3.45980930104479771241452177707, −2.14553287380922104654194180223, −1.67599868090386320027525555506,
1.67599868090386320027525555506, 2.14553287380922104654194180223, 3.45980930104479771241452177707, 4.27595667273238433856864492537, 4.88278545626254278974863093031, 5.53655916663109232849192235123, 6.30562534794010891274807107274, 7.14771969117923871254393122366, 7.80508399651374709383839404674, 8.776827255747300284110583645747
