| L(s) = 1 | − 2.18·2-s − 3-s + 2.76·4-s + 5-s + 2.18·6-s − 1.44·7-s − 1.66·8-s + 9-s − 2.18·10-s − 3.50·11-s − 2.76·12-s + 13-s + 3.15·14-s − 15-s − 1.89·16-s + 3.69·17-s − 2.18·18-s + 3.16·19-s + 2.76·20-s + 1.44·21-s + 7.65·22-s − 2.36·23-s + 1.66·24-s + 25-s − 2.18·26-s − 27-s − 3.99·28-s + ⋯ |
| L(s) = 1 | − 1.54·2-s − 0.577·3-s + 1.38·4-s + 0.447·5-s + 0.890·6-s − 0.547·7-s − 0.587·8-s + 0.333·9-s − 0.690·10-s − 1.05·11-s − 0.797·12-s + 0.277·13-s + 0.844·14-s − 0.258·15-s − 0.473·16-s + 0.896·17-s − 0.514·18-s + 0.726·19-s + 0.617·20-s + 0.315·21-s + 1.63·22-s − 0.492·23-s + 0.339·24-s + 0.200·25-s − 0.427·26-s − 0.192·27-s − 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5502724050\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5502724050\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 2.18T + 2T^{2} \) |
| 7 | \( 1 + 1.44T + 7T^{2} \) |
| 11 | \( 1 + 3.50T + 11T^{2} \) |
| 17 | \( 1 - 3.69T + 17T^{2} \) |
| 19 | \( 1 - 3.16T + 19T^{2} \) |
| 23 | \( 1 + 2.36T + 23T^{2} \) |
| 29 | \( 1 + 2.80T + 29T^{2} \) |
| 37 | \( 1 + 3.36T + 37T^{2} \) |
| 41 | \( 1 + 2.08T + 41T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 - 2.96T + 47T^{2} \) |
| 53 | \( 1 + 6.88T + 53T^{2} \) |
| 59 | \( 1 - 12.8T + 59T^{2} \) |
| 61 | \( 1 + 0.163T + 61T^{2} \) |
| 67 | \( 1 - 1.56T + 67T^{2} \) |
| 71 | \( 1 + 8.10T + 71T^{2} \) |
| 73 | \( 1 + 4.95T + 73T^{2} \) |
| 79 | \( 1 + 7.77T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 - 3.52T + 89T^{2} \) |
| 97 | \( 1 - 7.93T + 97T^{2} \) |
| 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
−8.070246447062033158214596505384, −7.47925071098784069956445226868, −6.93858227706619556929215145274, −5.97735183009980047999833287420, −5.55074401873210008082032728205, −4.57312973654118578932613484147, −3.38008052120421896868264477776, −2.46564022236373692088835496914, −1.49009234469761469728471289618, −0.52950523016932681724982374496,
0.52950523016932681724982374496, 1.49009234469761469728471289618, 2.46564022236373692088835496914, 3.38008052120421896868264477776, 4.57312973654118578932613484147, 5.55074401873210008082032728205, 5.97735183009980047999833287420, 6.93858227706619556929215145274, 7.47925071098784069956445226868, 8.070246447062033158214596505384
