| L(s) = 1 | − 2.57·2-s + 4.64·4-s + 2.12·5-s − 3.45·7-s − 6.80·8-s − 5.48·10-s − 11-s − 2.79·13-s + 8.90·14-s + 8.25·16-s − 6.16·17-s − 2.59·19-s + 9.86·20-s + 2.57·22-s + 0.281·23-s − 0.476·25-s + 7.20·26-s − 16.0·28-s − 0.776·29-s + 3.56·31-s − 7.65·32-s + 15.8·34-s − 7.34·35-s − 11.5·37-s + 6.68·38-s − 14.4·40-s + 10.0·41-s + ⋯ |
| L(s) = 1 | − 1.82·2-s + 2.32·4-s + 0.951·5-s − 1.30·7-s − 2.40·8-s − 1.73·10-s − 0.301·11-s − 0.775·13-s + 2.37·14-s + 2.06·16-s − 1.49·17-s − 0.594·19-s + 2.20·20-s + 0.549·22-s + 0.0587·23-s − 0.0952·25-s + 1.41·26-s − 3.02·28-s − 0.144·29-s + 0.640·31-s − 1.35·32-s + 2.72·34-s − 1.24·35-s − 1.89·37-s + 1.08·38-s − 2.28·40-s + 1.57·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3455495015\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3455495015\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 + 2.57T + 2T^{2} \) |
| 5 | \( 1 - 2.12T + 5T^{2} \) |
| 7 | \( 1 + 3.45T + 7T^{2} \) |
| 13 | \( 1 + 2.79T + 13T^{2} \) |
| 17 | \( 1 + 6.16T + 17T^{2} \) |
| 19 | \( 1 + 2.59T + 19T^{2} \) |
| 23 | \( 1 - 0.281T + 23T^{2} \) |
| 29 | \( 1 + 0.776T + 29T^{2} \) |
| 31 | \( 1 - 3.56T + 31T^{2} \) |
| 37 | \( 1 + 11.5T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 + 4.63T + 43T^{2} \) |
| 47 | \( 1 + 0.572T + 47T^{2} \) |
| 53 | \( 1 + 1.09T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 67 | \( 1 + 11.7T + 67T^{2} \) |
| 71 | \( 1 - 0.0926T + 71T^{2} \) |
| 73 | \( 1 + 3.82T + 73T^{2} \) |
| 79 | \( 1 + 8.59T + 79T^{2} \) |
| 83 | \( 1 + 2.92T + 83T^{2} \) |
| 89 | \( 1 + 5.34T + 89T^{2} \) |
| 97 | \( 1 - 4.05T + 97T^{2} \) |
| 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
−8.366419364680063045174133412930, −7.30230462124964801311394590263, −6.87102648538422569499079268262, −6.27961439922045756192061198623, −5.64384901129639661163565610958, −4.43761512040295546742958783228, −3.12128583499778717097151075508, −2.38914157540894197375902394773, −1.80517823152107117471961529621, −0.38124884963106478224737584676,
0.38124884963106478224737584676, 1.80517823152107117471961529621, 2.38914157540894197375902394773, 3.12128583499778717097151075508, 4.43761512040295546742958783228, 5.64384901129639661163565610958, 6.27961439922045756192061198623, 6.87102648538422569499079268262, 7.30230462124964801311394590263, 8.366419364680063045174133412930
