| L(s) = 1 | − 0.712·2-s − 8.79·3-s − 31.4·4-s + 6.25·6-s + 199.·7-s + 45.2·8-s − 165.·9-s + 482.·11-s + 276.·12-s − 169.·13-s − 141.·14-s + 975.·16-s + 289·17-s + 117.·18-s − 2.18e3·19-s − 1.75e3·21-s − 343.·22-s + 4.07e3·23-s − 397.·24-s + 120.·26-s + 3.59e3·27-s − 6.28e3·28-s − 3.37e3·29-s − 4.33e3·31-s − 2.14e3·32-s − 4.23e3·33-s − 205.·34-s + ⋯ |
| L(s) = 1 | − 0.125·2-s − 0.563·3-s − 0.984·4-s + 0.0709·6-s + 1.53·7-s + 0.249·8-s − 0.681·9-s + 1.20·11-s + 0.555·12-s − 0.277·13-s − 0.193·14-s + 0.952·16-s + 0.242·17-s + 0.0858·18-s − 1.38·19-s − 0.867·21-s − 0.151·22-s + 1.60·23-s − 0.140·24-s + 0.0349·26-s + 0.948·27-s − 1.51·28-s − 0.746·29-s − 0.810·31-s − 0.369·32-s − 0.677·33-s − 0.0305·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.331444990\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.331444990\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 - 289T \) |
| good | 2 | \( 1 + 0.712T + 32T^{2} \) |
| 3 | \( 1 + 8.79T + 243T^{2} \) |
| 7 | \( 1 - 199.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 482.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 169.T + 3.71e5T^{2} \) |
| 19 | \( 1 + 2.18e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.07e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.37e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.33e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 8.81e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.11e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.76e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 5.55e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.83e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.91e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.22e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.32e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.41e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.62e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.09e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.99e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.91e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.63e5T + 8.58e9T^{2} \) |
| 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
−10.59045235646969919278670470669, −9.191434664430081075623582669373, −8.699473983832040169749776097512, −7.79505927372330725358002426030, −6.54885934962240754270834917207, −5.31246025974598716815119017868, −4.76951815128273882441133507673, −3.65709228778607174652088069865, −1.79918528000547288981641364867, −0.66247451803709218126215074361,
0.66247451803709218126215074361, 1.79918528000547288981641364867, 3.65709228778607174652088069865, 4.76951815128273882441133507673, 5.31246025974598716815119017868, 6.54885934962240754270834917207, 7.79505927372330725358002426030, 8.699473983832040169749776097512, 9.191434664430081075623582669373, 10.59045235646969919278670470669
