| L(s) = 1 | + 2.26·2-s + 3-s + 3.10·4-s + 2.26·6-s + 4.96·7-s + 2.50·8-s + 9-s − 3.21·11-s + 3.10·12-s + 13-s + 11.2·14-s − 0.551·16-s − 0.448·17-s + 2.26·18-s − 7.23·19-s + 4.96·21-s − 7.27·22-s − 6.26·23-s + 2.50·24-s + 2.26·26-s + 27-s + 15.4·28-s − 2.21·29-s + 2.19·31-s − 6.26·32-s − 3.21·33-s − 1.01·34-s + ⋯ |
| L(s) = 1 | + 1.59·2-s + 0.577·3-s + 1.55·4-s + 0.922·6-s + 1.87·7-s + 0.886·8-s + 0.333·9-s − 0.969·11-s + 0.897·12-s + 0.277·13-s + 3.00·14-s − 0.137·16-s − 0.108·17-s + 0.532·18-s − 1.65·19-s + 1.08·21-s − 1.55·22-s − 1.30·23-s + 0.511·24-s + 0.443·26-s + 0.192·27-s + 2.92·28-s − 0.411·29-s + 0.393·31-s − 1.10·32-s − 0.559·33-s − 0.173·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.015363556\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.015363556\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 2.26T + 2T^{2} \) |
| 7 | \( 1 - 4.96T + 7T^{2} \) |
| 11 | \( 1 + 3.21T + 11T^{2} \) |
| 17 | \( 1 + 0.448T + 17T^{2} \) |
| 19 | \( 1 + 7.23T + 19T^{2} \) |
| 23 | \( 1 + 6.26T + 23T^{2} \) |
| 29 | \( 1 + 2.21T + 29T^{2} \) |
| 31 | \( 1 - 2.19T + 31T^{2} \) |
| 37 | \( 1 - 8.26T + 37T^{2} \) |
| 41 | \( 1 - 1.79T + 41T^{2} \) |
| 43 | \( 1 - 6.21T + 43T^{2} \) |
| 47 | \( 1 - 1.66T + 47T^{2} \) |
| 53 | \( 1 - 1.16T + 53T^{2} \) |
| 59 | \( 1 + 5.88T + 59T^{2} \) |
| 61 | \( 1 + 1.76T + 61T^{2} \) |
| 67 | \( 1 + 2.73T + 67T^{2} \) |
| 71 | \( 1 - 4.11T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 - 1.05T + 79T^{2} \) |
| 83 | \( 1 + 5.77T + 83T^{2} \) |
| 89 | \( 1 - 11.1T + 89T^{2} \) |
| 97 | \( 1 + 8.37T + 97T^{2} \) |
| 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
−10.41189507588337217996796173882, −8.950034403390619828596387854484, −8.089991863719147462779264694100, −7.56813889179980443817876519034, −6.26325994603940462878739002904, −5.46752313998264190599088019769, −4.47773186873243658779018809973, −4.12499076574557542646832450725, −2.62217685363767527875635859507, −1.91116546583365435344647393558,
1.91116546583365435344647393558, 2.62217685363767527875635859507, 4.12499076574557542646832450725, 4.47773186873243658779018809973, 5.46752313998264190599088019769, 6.26325994603940462878739002904, 7.56813889179980443817876519034, 8.089991863719147462779264694100, 8.950034403390619828596387854484, 10.41189507588337217996796173882
