| L(s) = 1 | − 2.49·2-s + 0.713·3-s + 4.20·4-s − 1.77·6-s + 3.20·7-s − 5.49·8-s − 2.49·9-s + 3.28·11-s + 2.99·12-s + 3.77·13-s − 7.98·14-s + 5.26·16-s − 0.981·17-s + 6.20·18-s + 4·19-s + 2.28·21-s − 8.18·22-s − 4·23-s − 3.91·24-s − 5·25-s − 9.40·26-s − 3.91·27-s + 13.4·28-s + 3.55·29-s + 2.40·31-s − 2.14·32-s + 2.34·33-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 0.411·3-s + 2.10·4-s − 0.725·6-s + 1.21·7-s − 1.94·8-s − 0.830·9-s + 0.990·11-s + 0.866·12-s + 1.04·13-s − 2.13·14-s + 1.31·16-s − 0.238·17-s + 1.46·18-s + 0.917·19-s + 0.498·21-s − 1.74·22-s − 0.834·23-s − 0.799·24-s − 25-s − 1.84·26-s − 0.754·27-s + 2.54·28-s + 0.660·29-s + 0.432·31-s − 0.378·32-s + 0.408·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8556943766\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8556943766\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 503 | \( 1 - T \) |
| good | 2 | \( 1 + 2.49T + 2T^{2} \) |
| 3 | \( 1 - 0.713T + 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 - 3.20T + 7T^{2} \) |
| 11 | \( 1 - 3.28T + 11T^{2} \) |
| 13 | \( 1 - 3.77T + 13T^{2} \) |
| 17 | \( 1 + 0.981T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 3.55T + 29T^{2} \) |
| 31 | \( 1 - 2.40T + 31T^{2} \) |
| 37 | \( 1 + 0.854T + 37T^{2} \) |
| 41 | \( 1 + 8.53T + 41T^{2} \) |
| 43 | \( 1 - 9.89T + 43T^{2} \) |
| 47 | \( 1 - 8.06T + 47T^{2} \) |
| 53 | \( 1 - 8.12T + 53T^{2} \) |
| 59 | \( 1 - 0.0637T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 14.9T + 67T^{2} \) |
| 71 | \( 1 - 8.98T + 71T^{2} \) |
| 73 | \( 1 - 6.30T + 73T^{2} \) |
| 79 | \( 1 + 7.90T + 79T^{2} \) |
| 83 | \( 1 + 3.20T + 83T^{2} \) |
| 89 | \( 1 + 2.12T + 89T^{2} \) |
| 97 | \( 1 + 17.0T + 97T^{2} \) |
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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
−10.83841372254274468191701638071, −9.810150190624472395826705438685, −8.965226817018298697378078685596, −8.355564340313463925359138371172, −7.81195689875417512458495789578, −6.68075267731393162194784248717, −5.62561880297439465388462578826, −3.87540961262458159797339684452, −2.30777564194275348621960509992, −1.17100951925839039355875305040,
1.17100951925839039355875305040, 2.30777564194275348621960509992, 3.87540961262458159797339684452, 5.62561880297439465388462578826, 6.68075267731393162194784248717, 7.81195689875417512458495789578, 8.355564340313463925359138371172, 8.965226817018298697378078685596, 9.810150190624472395826705438685, 10.83841372254274468191701638071
