| L(s) = 1 | − 2.53·2-s + 0.289·3-s + 4.40·4-s − 2.52·5-s − 0.731·6-s + 7-s − 6.09·8-s − 2.91·9-s + 6.39·10-s + 3.22·11-s + 1.27·12-s + 0.295·13-s − 2.53·14-s − 0.730·15-s + 6.60·16-s − 1.30·17-s + 7.38·18-s − 5.37·19-s − 11.1·20-s + 0.289·21-s − 8.15·22-s + 6.32·23-s − 1.76·24-s + 1.38·25-s − 0.747·26-s − 1.71·27-s + 4.40·28-s + ⋯ |
| L(s) = 1 | − 1.78·2-s + 0.166·3-s + 2.20·4-s − 1.12·5-s − 0.298·6-s + 0.377·7-s − 2.15·8-s − 0.972·9-s + 2.02·10-s + 0.971·11-s + 0.367·12-s + 0.0819·13-s − 0.676·14-s − 0.188·15-s + 1.65·16-s − 0.317·17-s + 1.73·18-s − 1.23·19-s − 2.48·20-s + 0.0630·21-s − 1.73·22-s + 1.31·23-s − 0.359·24-s + 0.276·25-s − 0.146·26-s − 0.329·27-s + 0.832·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4391562966\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4391562966\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 859 | \( 1 + T \) |
| good | 2 | \( 1 + 2.53T + 2T^{2} \) |
| 3 | \( 1 - 0.289T + 3T^{2} \) |
| 5 | \( 1 + 2.52T + 5T^{2} \) |
| 11 | \( 1 - 3.22T + 11T^{2} \) |
| 13 | \( 1 - 0.295T + 13T^{2} \) |
| 17 | \( 1 + 1.30T + 17T^{2} \) |
| 19 | \( 1 + 5.37T + 19T^{2} \) |
| 23 | \( 1 - 6.32T + 23T^{2} \) |
| 29 | \( 1 + 1.72T + 29T^{2} \) |
| 31 | \( 1 - 5.03T + 31T^{2} \) |
| 37 | \( 1 + 10.0T + 37T^{2} \) |
| 41 | \( 1 + 6.07T + 41T^{2} \) |
| 43 | \( 1 + 7.48T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 - 3.83T + 53T^{2} \) |
| 59 | \( 1 - 1.17T + 59T^{2} \) |
| 61 | \( 1 - 14.4T + 61T^{2} \) |
| 67 | \( 1 + 8.85T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 83 | \( 1 - 1.56T + 83T^{2} \) |
| 89 | \( 1 - 0.986T + 89T^{2} \) |
| 97 | \( 1 + 1.18T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.454796209775239500417818123976, −7.57052237232873618433484415062, −6.92604841503680291460333839619, −6.44583033542692753525096357849, −5.35567174737158897765339361187, −4.25802927768361875384727537140, −3.42670572737171075315236860857, −2.50408030911035879891936045569, −1.55222894237903062376361814938, −0.45984473904386272801815398866,
0.45984473904386272801815398866, 1.55222894237903062376361814938, 2.50408030911035879891936045569, 3.42670572737171075315236860857, 4.25802927768361875384727537140, 5.35567174737158897765339361187, 6.44583033542692753525096357849, 6.92604841503680291460333839619, 7.57052237232873618433484415062, 8.454796209775239500417818123976
