| L(s) = 1 | − 2.61·2-s − 2.23·3-s + 4.85·4-s + 2.23·5-s + 5.85·6-s − 7.47·8-s + 2.00·9-s − 5.85·10-s − 3·11-s − 10.8·12-s − 13-s − 5.00·15-s + 9.85·16-s + 1.47·17-s − 5.23·18-s + 3·19-s + 10.8·20-s + 7.85·22-s − 8.23·23-s + 16.7·24-s + 2.61·26-s + 2.23·27-s + 4.47·29-s + 13.0·30-s + 5·31-s − 10.8·32-s + 6.70·33-s + ⋯ |
| L(s) = 1 | − 1.85·2-s − 1.29·3-s + 2.42·4-s + 0.999·5-s + 2.38·6-s − 2.64·8-s + 0.666·9-s − 1.85·10-s − 0.904·11-s − 3.13·12-s − 0.277·13-s − 1.29·15-s + 2.46·16-s + 0.357·17-s − 1.23·18-s + 0.688·19-s + 2.42·20-s + 1.67·22-s − 1.71·23-s + 3.41·24-s + 0.513·26-s + 0.430·27-s + 0.830·29-s + 2.38·30-s + 0.898·31-s − 1.91·32-s + 1.16·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 2.61T + 2T^{2} \) |
| 3 | \( 1 + 2.23T + 3T^{2} \) |
| 5 | \( 1 - 2.23T + 5T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 17 | \( 1 - 1.47T + 17T^{2} \) |
| 19 | \( 1 - 3T + 19T^{2} \) |
| 23 | \( 1 + 8.23T + 23T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 - 4.70T + 37T^{2} \) |
| 41 | \( 1 + 4.47T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 7.47T + 47T^{2} \) |
| 53 | \( 1 + 7.47T + 53T^{2} \) |
| 59 | \( 1 + 1.47T + 59T^{2} \) |
| 61 | \( 1 - 3T + 61T^{2} \) |
| 67 | \( 1 + 3T + 67T^{2} \) |
| 71 | \( 1 + 8.94T + 71T^{2} \) |
| 73 | \( 1 + 2.70T + 73T^{2} \) |
| 79 | \( 1 + 2.70T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 2.23T + 89T^{2} \) |
| 97 | \( 1 - 9.41T + 97T^{2} \) |
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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
−10.04739722696444408909984293125, −9.696184314450238948679944456992, −8.391231870394862979874925174825, −7.69159979186934493867487892216, −6.53198305530027523178787819232, −6.00430706481802488805935210728, −5.05312515101241452638497988149, −2.72343313377748065226121554450, −1.48329437481324086898099118124, 0,
1.48329437481324086898099118124, 2.72343313377748065226121554450, 5.05312515101241452638497988149, 6.00430706481802488805935210728, 6.53198305530027523178787819232, 7.69159979186934493867487892216, 8.391231870394862979874925174825, 9.696184314450238948679944456992, 10.04739722696444408909984293125
