| L(s) = 1 | − 2.29·2-s + 3.25·4-s − 2.24·5-s − 1.95·7-s − 2.87·8-s + 5.13·10-s + 2.64·11-s + 1.14·13-s + 4.48·14-s + 0.0755·16-s + 0.776·17-s + 6.71·19-s − 7.29·20-s − 6.06·22-s − 6.05·23-s + 0.0277·25-s − 2.61·26-s − 6.36·28-s − 6.09·29-s + 6.24·31-s + 5.56·32-s − 1.78·34-s + 4.38·35-s + 2.65·37-s − 15.3·38-s + 6.43·40-s − 10.9·41-s + ⋯ |
| L(s) = 1 | − 1.62·2-s + 1.62·4-s − 1.00·5-s − 0.739·7-s − 1.01·8-s + 1.62·10-s + 0.797·11-s + 0.316·13-s + 1.19·14-s + 0.0188·16-s + 0.188·17-s + 1.54·19-s − 1.63·20-s − 1.29·22-s − 1.26·23-s + 0.00555·25-s − 0.512·26-s − 1.20·28-s − 1.13·29-s + 1.12·31-s + 0.984·32-s − 0.305·34-s + 0.741·35-s + 0.437·37-s − 2.49·38-s + 1.01·40-s − 1.70·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 223 | \( 1 + T \) |
| good | 2 | \( 1 + 2.29T + 2T^{2} \) |
| 5 | \( 1 + 2.24T + 5T^{2} \) |
| 7 | \( 1 + 1.95T + 7T^{2} \) |
| 11 | \( 1 - 2.64T + 11T^{2} \) |
| 13 | \( 1 - 1.14T + 13T^{2} \) |
| 17 | \( 1 - 0.776T + 17T^{2} \) |
| 19 | \( 1 - 6.71T + 19T^{2} \) |
| 23 | \( 1 + 6.05T + 23T^{2} \) |
| 29 | \( 1 + 6.09T + 29T^{2} \) |
| 31 | \( 1 - 6.24T + 31T^{2} \) |
| 37 | \( 1 - 2.65T + 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 + 0.535T + 43T^{2} \) |
| 47 | \( 1 - 7.95T + 47T^{2} \) |
| 53 | \( 1 + 9.04T + 53T^{2} \) |
| 59 | \( 1 + 9.09T + 59T^{2} \) |
| 61 | \( 1 + 8.51T + 61T^{2} \) |
| 67 | \( 1 - 11.8T + 67T^{2} \) |
| 71 | \( 1 - 8.92T + 71T^{2} \) |
| 73 | \( 1 + 0.953T + 73T^{2} \) |
| 79 | \( 1 - 13.2T + 79T^{2} \) |
| 83 | \( 1 - 5.37T + 83T^{2} \) |
| 89 | \( 1 + 6.40T + 89T^{2} \) |
| 97 | \( 1 + 8.35T + 97T^{2} \) |
| 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
−7.83821844631923175731650193966, −7.34863185565654950359346272435, −6.55901716964293362194913473558, −5.98796433458212119894667633622, −4.76646913523386362256780575874, −3.74078397672997009560105333222, −3.22512264328485209736036254281, −1.96212280090292268994203629731, −0.979843295214325606836958275227, 0,
0.979843295214325606836958275227, 1.96212280090292268994203629731, 3.22512264328485209736036254281, 3.74078397672997009560105333222, 4.76646913523386362256780575874, 5.98796433458212119894667633622, 6.55901716964293362194913473558, 7.34863185565654950359346272435, 7.83821844631923175731650193966
