| L(s) = 1 | + 1.39·2-s − 3-s − 0.0647·4-s + 5-s − 1.39·6-s + 1.38·7-s − 2.87·8-s + 9-s + 1.39·10-s + 5.69·11-s + 0.0647·12-s + 13-s + 1.92·14-s − 15-s − 3.86·16-s − 1.02·17-s + 1.39·18-s − 2.05·19-s − 0.0647·20-s − 1.38·21-s + 7.92·22-s − 0.355·23-s + 2.87·24-s + 25-s + 1.39·26-s − 27-s − 0.0895·28-s + ⋯ |
| L(s) = 1 | + 0.983·2-s − 0.577·3-s − 0.0323·4-s + 0.447·5-s − 0.567·6-s + 0.522·7-s − 1.01·8-s + 0.333·9-s + 0.439·10-s + 1.71·11-s + 0.0186·12-s + 0.277·13-s + 0.514·14-s − 0.258·15-s − 0.966·16-s − 0.249·17-s + 0.327·18-s − 0.471·19-s − 0.0144·20-s − 0.301·21-s + 1.69·22-s − 0.0741·23-s + 0.586·24-s + 0.200·25-s + 0.272·26-s − 0.192·27-s − 0.0169·28-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
| \(L(1)\) |
\(\approx\) |
\(3.070403567\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.070403567\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.39T + 2T^{2} \) |
| 7 | \( 1 - 1.38T + 7T^{2} \) |
| 11 | \( 1 - 5.69T + 11T^{2} \) |
| 17 | \( 1 + 1.02T + 17T^{2} \) |
| 19 | \( 1 + 2.05T + 19T^{2} \) |
| 23 | \( 1 + 0.355T + 23T^{2} \) |
| 29 | \( 1 - 8.85T + 29T^{2} \) |
| 37 | \( 1 - 9.22T + 37T^{2} \) |
| 41 | \( 1 + 8.53T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 + 1.95T + 53T^{2} \) |
| 59 | \( 1 - 5.78T + 59T^{2} \) |
| 61 | \( 1 - 1.90T + 61T^{2} \) |
| 67 | \( 1 + 5.58T + 67T^{2} \) |
| 71 | \( 1 + 0.557T + 71T^{2} \) |
| 73 | \( 1 - 3.48T + 73T^{2} \) |
| 79 | \( 1 - 4.77T + 79T^{2} \) |
| 83 | \( 1 - 7.59T + 83T^{2} \) |
| 89 | \( 1 + 7.46T + 89T^{2} \) |
| 97 | \( 1 + 0.592T + 97T^{2} \) |
| 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
−8.223889366380268470678203293864, −6.88277799264992802900769096861, −6.47685378856073979471488759550, −5.93707849836379278590539014765, −5.08514579486862506279265021537, −4.47835349899894938497512974809, −3.94449843837066629268666920047, −3.00636463401064095866172784056, −1.85355419407338701401067460136, −0.858147058574460298084730185316,
0.858147058574460298084730185316, 1.85355419407338701401067460136, 3.00636463401064095866172784056, 3.94449843837066629268666920047, 4.47835349899894938497512974809, 5.08514579486862506279265021537, 5.93707849836379278590539014765, 6.47685378856073979471488759550, 6.88277799264992802900769096861, 8.223889366380268470678203293864
