| L(s) = 1 | − 2.64·2-s − 0.0865·3-s + 5.00·4-s − 1.46·5-s + 0.229·6-s − 4.80·7-s − 7.94·8-s − 2.99·9-s + 3.87·10-s + 2.57·11-s − 0.432·12-s + 4.36·13-s + 12.7·14-s + 0.126·15-s + 11.0·16-s + 5.06·17-s + 7.91·18-s + 3.86·19-s − 7.33·20-s + 0.416·21-s − 6.81·22-s + 5.72·23-s + 0.687·24-s − 2.84·25-s − 11.5·26-s + 0.518·27-s − 24.0·28-s + ⋯ |
| L(s) = 1 | − 1.87·2-s − 0.0499·3-s + 2.50·4-s − 0.655·5-s + 0.0935·6-s − 1.81·7-s − 2.80·8-s − 0.997·9-s + 1.22·10-s + 0.776·11-s − 0.124·12-s + 1.21·13-s + 3.39·14-s + 0.0327·15-s + 2.75·16-s + 1.22·17-s + 1.86·18-s + 0.887·19-s − 1.63·20-s + 0.0908·21-s − 1.45·22-s + 1.19·23-s + 0.140·24-s − 0.569·25-s − 2.26·26-s + 0.0998·27-s − 4.54·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5645762006\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5645762006\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 8011 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.64T + 2T^{2} \) |
| 3 | \( 1 + 0.0865T + 3T^{2} \) |
| 5 | \( 1 + 1.46T + 5T^{2} \) |
| 7 | \( 1 + 4.80T + 7T^{2} \) |
| 11 | \( 1 - 2.57T + 11T^{2} \) |
| 13 | \( 1 - 4.36T + 13T^{2} \) |
| 17 | \( 1 - 5.06T + 17T^{2} \) |
| 19 | \( 1 - 3.86T + 19T^{2} \) |
| 23 | \( 1 - 5.72T + 23T^{2} \) |
| 29 | \( 1 - 4.16T + 29T^{2} \) |
| 31 | \( 1 - 1.73T + 31T^{2} \) |
| 37 | \( 1 - 6.26T + 37T^{2} \) |
| 41 | \( 1 + 5.65T + 41T^{2} \) |
| 43 | \( 1 - 3.61T + 43T^{2} \) |
| 47 | \( 1 + 3.93T + 47T^{2} \) |
| 53 | \( 1 + 0.0159T + 53T^{2} \) |
| 59 | \( 1 - 9.52T + 59T^{2} \) |
| 61 | \( 1 + 5.30T + 61T^{2} \) |
| 67 | \( 1 - 6.24T + 67T^{2} \) |
| 71 | \( 1 - 3.50T + 71T^{2} \) |
| 73 | \( 1 - 7.45T + 73T^{2} \) |
| 79 | \( 1 - 5.44T + 79T^{2} \) |
| 83 | \( 1 + 15.8T + 83T^{2} \) |
| 89 | \( 1 - 4.26T + 89T^{2} \) |
| 97 | \( 1 + 9.83T + 97T^{2} \) |
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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
−8.081135875707584553948513489152, −7.24812224989042388712295812657, −6.63443579214023409031560232913, −6.16846288085905713557835654045, −5.48653498976529945212325062458, −3.77366547205643367044958199497, −3.23934744108468757692951810939, −2.69737160625381355761389144109, −1.18809646972091571288938974485, −0.58182875233866720010268967045,
0.58182875233866720010268967045, 1.18809646972091571288938974485, 2.69737160625381355761389144109, 3.23934744108468757692951810939, 3.77366547205643367044958199497, 5.48653498976529945212325062458, 6.16846288085905713557835654045, 6.63443579214023409031560232913, 7.24812224989042388712295812657, 8.081135875707584553948513489152
