| L(s) = 1 | + 1.86·2-s − 2.01·3-s + 1.48·4-s − 3.75·6-s + 2.04·7-s − 0.966·8-s + 1.05·9-s + 11-s − 2.98·12-s + 4.91·13-s + 3.80·14-s − 4.76·16-s + 6.79·17-s + 1.96·18-s − 19-s − 4.10·21-s + 1.86·22-s − 1.43·23-s + 1.94·24-s + 9.17·26-s + 3.92·27-s + 3.02·28-s − 0.704·29-s − 4.21·31-s − 6.96·32-s − 2.01·33-s + 12.6·34-s + ⋯ |
| L(s) = 1 | + 1.31·2-s − 1.16·3-s + 0.740·4-s − 1.53·6-s + 0.771·7-s − 0.341·8-s + 0.350·9-s + 0.301·11-s − 0.861·12-s + 1.36·13-s + 1.01·14-s − 1.19·16-s + 1.64·17-s + 0.462·18-s − 0.229·19-s − 0.896·21-s + 0.397·22-s − 0.299·23-s + 0.397·24-s + 1.79·26-s + 0.754·27-s + 0.571·28-s − 0.130·29-s − 0.757·31-s − 1.23·32-s − 0.350·33-s + 2.17·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.848405001\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.848405001\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 1.86T + 2T^{2} \) |
| 3 | \( 1 + 2.01T + 3T^{2} \) |
| 7 | \( 1 - 2.04T + 7T^{2} \) |
| 13 | \( 1 - 4.91T + 13T^{2} \) |
| 17 | \( 1 - 6.79T + 17T^{2} \) |
| 23 | \( 1 + 1.43T + 23T^{2} \) |
| 29 | \( 1 + 0.704T + 29T^{2} \) |
| 31 | \( 1 + 4.21T + 31T^{2} \) |
| 37 | \( 1 - 1.40T + 37T^{2} \) |
| 41 | \( 1 + 9.38T + 41T^{2} \) |
| 43 | \( 1 + 3.80T + 43T^{2} \) |
| 47 | \( 1 - 7.13T + 47T^{2} \) |
| 53 | \( 1 + 2.65T + 53T^{2} \) |
| 59 | \( 1 - 13.3T + 59T^{2} \) |
| 61 | \( 1 - 7.58T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 - 5.75T + 71T^{2} \) |
| 73 | \( 1 - 5.49T + 73T^{2} \) |
| 79 | \( 1 - 7.12T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 + 3.26T + 89T^{2} \) |
| 97 | \( 1 - 1.64T + 97T^{2} \) |
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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
−8.182123282724572340028169142246, −7.14846519208835673312995527897, −6.36631931103160128276228688596, −5.82003669852222790076131816362, −5.32358604559115043082497938407, −4.73893335730636834167308225758, −3.79529993743309114701897751450, −3.30748684563927528427101971116, −1.90905657450976187260176699490, −0.820962561041616855402232198411,
0.820962561041616855402232198411, 1.90905657450976187260176699490, 3.30748684563927528427101971116, 3.79529993743309114701897751450, 4.73893335730636834167308225758, 5.32358604559115043082497938407, 5.82003669852222790076131816362, 6.36631931103160128276228688596, 7.14846519208835673312995527897, 8.182123282724572340028169142246
