| L(s) = 1 | − 2.64·2-s + 3.39·3-s + 4.98·4-s − 0.538·5-s − 8.98·6-s − 0.766·7-s − 7.89·8-s + 8.54·9-s + 1.42·10-s + 3.53·11-s + 16.9·12-s + 5.95·13-s + 2.02·14-s − 1.83·15-s + 10.8·16-s + 4.30·17-s − 22.5·18-s − 3.03·19-s − 2.68·20-s − 2.60·21-s − 9.33·22-s − 0.904·23-s − 26.8·24-s − 4.70·25-s − 15.7·26-s + 18.8·27-s − 3.82·28-s + ⋯ |
| L(s) = 1 | − 1.86·2-s + 1.96·3-s + 2.49·4-s − 0.241·5-s − 3.66·6-s − 0.289·7-s − 2.78·8-s + 2.84·9-s + 0.450·10-s + 1.06·11-s + 4.89·12-s + 1.65·13-s + 0.541·14-s − 0.472·15-s + 2.72·16-s + 1.04·17-s − 5.32·18-s − 0.696·19-s − 0.600·20-s − 0.568·21-s − 1.98·22-s − 0.188·23-s − 5.47·24-s − 0.941·25-s − 3.08·26-s + 3.62·27-s − 0.722·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.056789975\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.056789975\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 4027 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.64T + 2T^{2} \) |
| 3 | \( 1 - 3.39T + 3T^{2} \) |
| 5 | \( 1 + 0.538T + 5T^{2} \) |
| 7 | \( 1 + 0.766T + 7T^{2} \) |
| 11 | \( 1 - 3.53T + 11T^{2} \) |
| 13 | \( 1 - 5.95T + 13T^{2} \) |
| 17 | \( 1 - 4.30T + 17T^{2} \) |
| 19 | \( 1 + 3.03T + 19T^{2} \) |
| 23 | \( 1 + 0.904T + 23T^{2} \) |
| 29 | \( 1 - 6.07T + 29T^{2} \) |
| 31 | \( 1 + 4.27T + 31T^{2} \) |
| 37 | \( 1 - 2.39T + 37T^{2} \) |
| 41 | \( 1 + 1.40T + 41T^{2} \) |
| 43 | \( 1 - 1.33T + 43T^{2} \) |
| 47 | \( 1 + 3.59T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 - 4.29T + 59T^{2} \) |
| 61 | \( 1 - 7.02T + 61T^{2} \) |
| 67 | \( 1 + 13.8T + 67T^{2} \) |
| 71 | \( 1 - 6.80T + 71T^{2} \) |
| 73 | \( 1 - 3.02T + 73T^{2} \) |
| 79 | \( 1 + 6.52T + 79T^{2} \) |
| 83 | \( 1 + 11.5T + 83T^{2} \) |
| 89 | \( 1 + 4.54T + 89T^{2} \) |
| 97 | \( 1 + 9.25T + 97T^{2} \) |
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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
−8.459270683195584944076159541693, −8.171989006793237844031752310422, −7.34315948373842658738558523356, −6.74573493269431644694795350785, −6.01666450722089582948502638141, −4.05718839602185197507878590889, −3.54556506121621411404436657108, −2.69931498082467136259688790850, −1.70083704789955077380277021793, −1.10884126268856288595198363443,
1.10884126268856288595198363443, 1.70083704789955077380277021793, 2.69931498082467136259688790850, 3.54556506121621411404436657108, 4.05718839602185197507878590889, 6.01666450722089582948502638141, 6.74573493269431644694795350785, 7.34315948373842658738558523356, 8.171989006793237844031752310422, 8.459270683195584944076159541693
