| L(s) = 1 | − 5.44·2-s − 3.90·3-s + 21.6·4-s − 5·5-s + 21.2·6-s + 26.4·7-s − 74.2·8-s − 11.7·9-s + 27.2·10-s + 11·11-s − 84.5·12-s + 4.93·13-s − 144.·14-s + 19.5·15-s + 230.·16-s + 84.7·17-s + 63.8·18-s − 19·19-s − 108.·20-s − 103.·21-s − 59.8·22-s − 50.4·23-s + 290.·24-s + 25·25-s − 26.8·26-s + 151.·27-s + 572.·28-s + ⋯ |
| L(s) = 1 | − 1.92·2-s − 0.752·3-s + 2.70·4-s − 0.447·5-s + 1.44·6-s + 1.42·7-s − 3.27·8-s − 0.434·9-s + 0.860·10-s + 0.301·11-s − 2.03·12-s + 0.105·13-s − 2.74·14-s + 0.336·15-s + 3.60·16-s + 1.20·17-s + 0.835·18-s − 0.229·19-s − 1.20·20-s − 1.07·21-s − 0.580·22-s − 0.456·23-s + 2.46·24-s + 0.200·25-s − 0.202·26-s + 1.07·27-s + 3.86·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 11 | \( 1 - 11T \) |
| 19 | \( 1 + 19T \) |
| good | 2 | \( 1 + 5.44T + 8T^{2} \) |
| 3 | \( 1 + 3.90T + 27T^{2} \) |
| 7 | \( 1 - 26.4T + 343T^{2} \) |
| 13 | \( 1 - 4.93T + 2.19e3T^{2} \) |
| 17 | \( 1 - 84.7T + 4.91e3T^{2} \) |
| 23 | \( 1 + 50.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 94.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 66.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 364.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 348.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 372.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 444.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 485.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 652.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 102.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 248.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 163.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 449.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 327.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 468.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 787.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.42e3T + 9.12e5T^{2} \) |
| 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
−8.956135648352365781846047987712, −8.319655784771671696413044726467, −7.74581346130040895091546481729, −6.94106262806638172667739850100, −5.93033529163983984170787657312, −5.09550886675973021105416452962, −3.44039728522773456964151355107, −1.99602456219688298373910027326, −1.10122325184705759383685805089, 0,
1.10122325184705759383685805089, 1.99602456219688298373910027326, 3.44039728522773456964151355107, 5.09550886675973021105416452962, 5.93033529163983984170787657312, 6.94106262806638172667739850100, 7.74581346130040895091546481729, 8.319655784771671696413044726467, 8.956135648352365781846047987712
