| L(s) = 1 | − 1.59·2-s + 0.551·4-s + 2.02·5-s − 1.90·7-s + 2.31·8-s − 3.23·10-s − 6.42·11-s + 0.315·13-s + 3.04·14-s − 4.79·16-s − 0.873·17-s − 4.25·19-s + 1.11·20-s + 10.2·22-s − 23-s − 0.908·25-s − 0.504·26-s − 1.05·28-s + 29-s + 2.71·31-s + 3.04·32-s + 1.39·34-s − 3.86·35-s − 8.06·37-s + 6.79·38-s + 4.67·40-s − 2.41·41-s + ⋯ |
| L(s) = 1 | − 1.12·2-s + 0.275·4-s + 0.904·5-s − 0.721·7-s + 0.817·8-s − 1.02·10-s − 1.93·11-s + 0.0875·13-s + 0.815·14-s − 1.19·16-s − 0.211·17-s − 0.975·19-s + 0.249·20-s + 2.18·22-s − 0.208·23-s − 0.181·25-s − 0.0988·26-s − 0.199·28-s + 0.185·29-s + 0.488·31-s + 0.537·32-s + 0.239·34-s − 0.652·35-s − 1.32·37-s + 1.10·38-s + 0.739·40-s − 0.377·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4801628659\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4801628659\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 1.59T + 2T^{2} \) |
| 5 | \( 1 - 2.02T + 5T^{2} \) |
| 7 | \( 1 + 1.90T + 7T^{2} \) |
| 11 | \( 1 + 6.42T + 11T^{2} \) |
| 13 | \( 1 - 0.315T + 13T^{2} \) |
| 17 | \( 1 + 0.873T + 17T^{2} \) |
| 19 | \( 1 + 4.25T + 19T^{2} \) |
| 31 | \( 1 - 2.71T + 31T^{2} \) |
| 37 | \( 1 + 8.06T + 37T^{2} \) |
| 41 | \( 1 + 2.41T + 41T^{2} \) |
| 43 | \( 1 + 1.70T + 43T^{2} \) |
| 47 | \( 1 + 6.56T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 - 9.53T + 67T^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 73 | \( 1 + 1.07T + 73T^{2} \) |
| 79 | \( 1 - 7.64T + 79T^{2} \) |
| 83 | \( 1 + 1.98T + 83T^{2} \) |
| 89 | \( 1 - 3.76T + 89T^{2} \) |
| 97 | \( 1 - 9.41T + 97T^{2} \) |
| 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
−8.325749126364923599647962870874, −7.52812577276285745482748758792, −6.79777733319556342190517044449, −6.07227701381597896563905647352, −5.26715867575557835061845253125, −4.62732929221167138865285196745, −3.44077640352736094074629997243, −2.42003098125959618631351770398, −1.84616468166506124934416933449, −0.41365794103337212167021414609,
0.41365794103337212167021414609, 1.84616468166506124934416933449, 2.42003098125959618631351770398, 3.44077640352736094074629997243, 4.62732929221167138865285196745, 5.26715867575557835061845253125, 6.07227701381597896563905647352, 6.79777733319556342190517044449, 7.52812577276285745482748758792, 8.325749126364923599647962870874
