| L(s) = 1 | + 2-s − 2.41·3-s + 4-s + 3.01·5-s − 2.41·6-s − 2.26·7-s + 8-s + 2.83·9-s + 3.01·10-s − 5.39·11-s − 2.41·12-s + 5.88·13-s − 2.26·14-s − 7.27·15-s + 16-s + 7.95·17-s + 2.83·18-s + 4.49·19-s + 3.01·20-s + 5.46·21-s − 5.39·22-s − 1.76·23-s − 2.41·24-s + 4.08·25-s + 5.88·26-s + 0.400·27-s − 2.26·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.39·3-s + 0.5·4-s + 1.34·5-s − 0.986·6-s − 0.855·7-s + 0.353·8-s + 0.944·9-s + 0.952·10-s − 1.62·11-s − 0.697·12-s + 1.63·13-s − 0.605·14-s − 1.87·15-s + 0.250·16-s + 1.93·17-s + 0.668·18-s + 1.03·19-s + 0.673·20-s + 1.19·21-s − 1.14·22-s − 0.367·23-s − 0.493·24-s + 0.816·25-s + 1.15·26-s + 0.0770·27-s − 0.427·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.477811944\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.477811944\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 4013 | \( 1+O(T) \) |
| good | 3 | \( 1 + 2.41T + 3T^{2} \) |
| 5 | \( 1 - 3.01T + 5T^{2} \) |
| 7 | \( 1 + 2.26T + 7T^{2} \) |
| 11 | \( 1 + 5.39T + 11T^{2} \) |
| 13 | \( 1 - 5.88T + 13T^{2} \) |
| 17 | \( 1 - 7.95T + 17T^{2} \) |
| 19 | \( 1 - 4.49T + 19T^{2} \) |
| 23 | \( 1 + 1.76T + 23T^{2} \) |
| 29 | \( 1 - 1.13T + 29T^{2} \) |
| 31 | \( 1 + 1.63T + 31T^{2} \) |
| 37 | \( 1 - 7.71T + 37T^{2} \) |
| 41 | \( 1 + 7.81T + 41T^{2} \) |
| 43 | \( 1 - 3.64T + 43T^{2} \) |
| 47 | \( 1 + 1.21T + 47T^{2} \) |
| 53 | \( 1 + 0.0872T + 53T^{2} \) |
| 59 | \( 1 - 1.78T + 59T^{2} \) |
| 61 | \( 1 + 3.26T + 61T^{2} \) |
| 67 | \( 1 + 1.83T + 67T^{2} \) |
| 71 | \( 1 + 0.645T + 71T^{2} \) |
| 73 | \( 1 + 4.00T + 73T^{2} \) |
| 79 | \( 1 + 17.5T + 79T^{2} \) |
| 83 | \( 1 - 11.9T + 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 - 3.74T + 97T^{2} \) |
| 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
−7.60500431141786943380731775727, −6.78517674458751794095725881445, −5.99173189657100261385027686339, −5.74013854175883895577122531498, −5.45951689467555615443779803825, −4.59592789963380632643010642888, −3.36080354249123814028115039070, −2.91947078366877410196031284205, −1.66053574311048216619321465555, −0.78106598536885363818767256289,
0.78106598536885363818767256289, 1.66053574311048216619321465555, 2.91947078366877410196031284205, 3.36080354249123814028115039070, 4.59592789963380632643010642888, 5.45951689467555615443779803825, 5.74013854175883895577122531498, 5.99173189657100261385027686339, 6.78517674458751794095725881445, 7.60500431141786943380731775727
