| L(s) = 1 | + 1.34·2-s − 0.184·4-s + 3.53·5-s − 2.53·7-s − 2.94·8-s + 4.75·10-s + 2.57·11-s + 3.94·13-s − 3.41·14-s − 3.59·16-s + 5.65·17-s − 0.532·19-s − 0.652·20-s + 3.46·22-s − 0.411·23-s + 7.47·25-s + 5.31·26-s + 0.467·28-s − 0.162·29-s + 9.29·31-s + 1.04·32-s + 7.61·34-s − 8.94·35-s + 5.43·37-s − 0.716·38-s − 10.3·40-s − 3.51·41-s + ⋯ |
| L(s) = 1 | + 0.952·2-s − 0.0923·4-s + 1.57·5-s − 0.957·7-s − 1.04·8-s + 1.50·10-s + 0.776·11-s + 1.09·13-s − 0.911·14-s − 0.899·16-s + 1.37·17-s − 0.122·19-s − 0.145·20-s + 0.739·22-s − 0.0857·23-s + 1.49·25-s + 1.04·26-s + 0.0884·28-s − 0.0301·29-s + 1.66·31-s + 0.184·32-s + 1.30·34-s − 1.51·35-s + 0.893·37-s − 0.116·38-s − 1.64·40-s − 0.549·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.985257751\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.985257751\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 - 1.34T + 2T^{2} \) |
| 5 | \( 1 - 3.53T + 5T^{2} \) |
| 7 | \( 1 + 2.53T + 7T^{2} \) |
| 11 | \( 1 - 2.57T + 11T^{2} \) |
| 13 | \( 1 - 3.94T + 13T^{2} \) |
| 17 | \( 1 - 5.65T + 17T^{2} \) |
| 19 | \( 1 + 0.532T + 19T^{2} \) |
| 23 | \( 1 + 0.411T + 23T^{2} \) |
| 29 | \( 1 + 0.162T + 29T^{2} \) |
| 31 | \( 1 - 9.29T + 31T^{2} \) |
| 37 | \( 1 - 5.43T + 37T^{2} \) |
| 41 | \( 1 + 3.51T + 41T^{2} \) |
| 43 | \( 1 + 12.1T + 43T^{2} \) |
| 47 | \( 1 - 6.14T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 + 2.57T + 59T^{2} \) |
| 61 | \( 1 + 12.9T + 61T^{2} \) |
| 67 | \( 1 - 0.532T + 67T^{2} \) |
| 71 | \( 1 + 6.84T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 - 0.568T + 79T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 + 15.2T + 97T^{2} \) |
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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
−9.839365280496257485466930772504, −9.156315633668147349523945484890, −8.396502150612938005985198821816, −6.81897748995771082853049540358, −6.03178540787566895065379386069, −5.83813426399427226935726206777, −4.66861037648709474466685992652, −3.56498678266320906164869455070, −2.81497325772301451659406931914, −1.28841828421406454033934078856,
1.28841828421406454033934078856, 2.81497325772301451659406931914, 3.56498678266320906164869455070, 4.66861037648709474466685992652, 5.83813426399427226935726206777, 6.03178540787566895065379386069, 6.81897748995771082853049540358, 8.396502150612938005985198821816, 9.156315633668147349523945484890, 9.839365280496257485466930772504
