| L(s) = 1 | − 0.565·2-s − 3-s − 1.67·4-s + 5-s + 0.565·6-s − 3.78·7-s + 2.08·8-s + 9-s − 0.565·10-s + 0.609·11-s + 1.67·12-s + 13-s + 2.14·14-s − 15-s + 2.18·16-s − 3.76·17-s − 0.565·18-s − 5.88·19-s − 1.67·20-s + 3.78·21-s − 0.344·22-s + 7.17·23-s − 2.08·24-s + 25-s − 0.565·26-s − 27-s + 6.36·28-s + ⋯ |
| L(s) = 1 | − 0.400·2-s − 0.577·3-s − 0.839·4-s + 0.447·5-s + 0.231·6-s − 1.43·7-s + 0.736·8-s + 0.333·9-s − 0.178·10-s + 0.183·11-s + 0.484·12-s + 0.277·13-s + 0.572·14-s − 0.258·15-s + 0.545·16-s − 0.913·17-s − 0.133·18-s − 1.34·19-s − 0.375·20-s + 0.826·21-s − 0.0734·22-s + 1.49·23-s − 0.425·24-s + 0.200·25-s − 0.110·26-s − 0.192·27-s + 1.20·28-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
| \(L(1)\) |
\(\approx\) |
\(0.4968513206\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4968513206\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.565T + 2T^{2} \) |
| 7 | \( 1 + 3.78T + 7T^{2} \) |
| 11 | \( 1 - 0.609T + 11T^{2} \) |
| 17 | \( 1 + 3.76T + 17T^{2} \) |
| 19 | \( 1 + 5.88T + 19T^{2} \) |
| 23 | \( 1 - 7.17T + 23T^{2} \) |
| 29 | \( 1 + 1.74T + 29T^{2} \) |
| 37 | \( 1 + 8.08T + 37T^{2} \) |
| 41 | \( 1 - 0.325T + 41T^{2} \) |
| 43 | \( 1 + 9.08T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 - 6.89T + 53T^{2} \) |
| 59 | \( 1 + 2.39T + 59T^{2} \) |
| 61 | \( 1 + 8.50T + 61T^{2} \) |
| 67 | \( 1 - 3.28T + 67T^{2} \) |
| 71 | \( 1 - 3.63T + 71T^{2} \) |
| 73 | \( 1 - 0.542T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 + 6.16T + 83T^{2} \) |
| 89 | \( 1 + 13.9T + 89T^{2} \) |
| 97 | \( 1 + 12.5T + 97T^{2} \) |
| 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
−8.328092080331532983161446448079, −7.07106246974536407700168324625, −6.76556560862948044700474952641, −5.97827508648035374890844582618, −5.27944241515225028398927303988, −4.43667517206641306898002443964, −3.76278733870905112878890487404, −2.80390740084797146716646080392, −1.59969280754881082155266387855, −0.41315389685437426677371962638,
0.41315389685437426677371962638, 1.59969280754881082155266387855, 2.80390740084797146716646080392, 3.76278733870905112878890487404, 4.43667517206641306898002443964, 5.27944241515225028398927303988, 5.97827508648035374890844582618, 6.76556560862948044700474952641, 7.07106246974536407700168324625, 8.328092080331532983161446448079
