| L(s) = 1 | − 2.74·2-s + 5.53·4-s − 1.53·5-s − 4.95·7-s − 9.70·8-s + 4.21·10-s + 2.42·11-s + 1.42·13-s + 13.6·14-s + 15.5·16-s − 17-s − 1.95·19-s − 8.48·20-s − 6.64·22-s + 6.06·23-s − 2.64·25-s − 3.90·26-s − 27.4·28-s + 1.53·29-s + 5.06·31-s − 23.3·32-s + 2.74·34-s + 7.60·35-s + 6.53·37-s + 5.36·38-s + 14.8·40-s + 1.53·41-s + ⋯ |
| L(s) = 1 | − 1.94·2-s + 2.76·4-s − 0.686·5-s − 1.87·7-s − 3.42·8-s + 1.33·10-s + 0.730·11-s + 0.394·13-s + 3.63·14-s + 3.88·16-s − 0.242·17-s − 0.448·19-s − 1.89·20-s − 1.41·22-s + 1.26·23-s − 0.529·25-s − 0.765·26-s − 5.18·28-s + 0.284·29-s + 0.910·31-s − 4.11·32-s + 0.470·34-s + 1.28·35-s + 1.07·37-s + 0.870·38-s + 2.35·40-s + 0.239·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3773126277\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3773126277\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 + 2.74T + 2T^{2} \) |
| 5 | \( 1 + 1.53T + 5T^{2} \) |
| 7 | \( 1 + 4.95T + 7T^{2} \) |
| 11 | \( 1 - 2.42T + 11T^{2} \) |
| 13 | \( 1 - 1.42T + 13T^{2} \) |
| 19 | \( 1 + 1.95T + 19T^{2} \) |
| 23 | \( 1 - 6.06T + 23T^{2} \) |
| 29 | \( 1 - 1.53T + 29T^{2} \) |
| 31 | \( 1 - 5.06T + 31T^{2} \) |
| 37 | \( 1 - 6.53T + 37T^{2} \) |
| 41 | \( 1 - 1.53T + 41T^{2} \) |
| 43 | \( 1 + 1.95T + 43T^{2} \) |
| 47 | \( 1 - 3.37T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 + 7.46T + 59T^{2} \) |
| 61 | \( 1 - 9.46T + 61T^{2} \) |
| 67 | \( 1 + 6.04T + 67T^{2} \) |
| 71 | \( 1 + 4.15T + 71T^{2} \) |
| 73 | \( 1 - 12.6T + 73T^{2} \) |
| 79 | \( 1 - 1.04T + 79T^{2} \) |
| 83 | \( 1 - 4.22T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 + 2.53T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.77067905828172048363762524460, −9.945248835387083263981073700990, −9.237189001060964757732725283271, −8.615691485136472301978960060125, −7.51716164479095069125246889980, −6.67902851926266528725714195859, −6.14490274548285443734616561804, −3.71300337165633073905240443527, −2.65575522992852392443889753074, −0.73358677570115489910869889713,
0.73358677570115489910869889713, 2.65575522992852392443889753074, 3.71300337165633073905240443527, 6.14490274548285443734616561804, 6.67902851926266528725714195859, 7.51716164479095069125246889980, 8.615691485136472301978960060125, 9.237189001060964757732725283271, 9.945248835387083263981073700990, 10.77067905828172048363762524460
