| L(s) = 1 | − 3.16·2-s − 8.60·3-s + 1.99·4-s + 5·5-s + 27.2·6-s + 32.5·7-s + 18.9·8-s + 47.1·9-s − 15.8·10-s − 11·11-s − 17.1·12-s + 59.8·13-s − 102.·14-s − 43.0·15-s − 75.9·16-s − 9.35·17-s − 148.·18-s − 19·19-s + 9.97·20-s − 280.·21-s + 34.7·22-s − 101.·23-s − 163.·24-s + 25·25-s − 189.·26-s − 173.·27-s + 64.9·28-s + ⋯ |
| L(s) = 1 | − 1.11·2-s − 1.65·3-s + 0.249·4-s + 0.447·5-s + 1.85·6-s + 1.75·7-s + 0.838·8-s + 1.74·9-s − 0.499·10-s − 0.301·11-s − 0.413·12-s + 1.27·13-s − 1.96·14-s − 0.740·15-s − 1.18·16-s − 0.133·17-s − 1.95·18-s − 0.229·19-s + 0.111·20-s − 2.91·21-s + 0.337·22-s − 0.919·23-s − 1.38·24-s + 0.200·25-s − 1.42·26-s − 1.23·27-s + 0.438·28-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 | \( 1 + 3.16T + 8T^{2} \) |
| 3 | \( 1 + 8.60T + 27T^{2} \) |
| 7 | \( 1 - 32.5T + 343T^{2} \) |
| 13 | \( 1 - 59.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 9.35T + 4.91e3T^{2} \) |
| 23 | \( 1 + 101.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 119.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 115.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 381.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 113.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 191.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 307.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 573.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 431.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 117.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 176.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 71.3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 323.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.32e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 284.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.14e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 630.T + 9.12e5T^{2} \) |
| 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
−9.119109009633642103886815729173, −8.289457801868769784867492914051, −7.64120188023371990980953353899, −6.57441742130605552208505421785, −5.68502693136323981637127934675, −4.94462138706979006883896729164, −4.18511092057559984590335385115, −1.78759104161297357822471660096, −1.22801773857144235037241033090, 0,
1.22801773857144235037241033090, 1.78759104161297357822471660096, 4.18511092057559984590335385115, 4.94462138706979006883896729164, 5.68502693136323981637127934675, 6.57441742130605552208505421785, 7.64120188023371990980953353899, 8.289457801868769784867492914051, 9.119109009633642103886815729173
