| L(s) = 1 | − 0.487·2-s − 1.76·4-s − 2.22·5-s − 0.937·7-s + 1.83·8-s + 1.08·10-s − 2.32·11-s − 4.97·13-s + 0.456·14-s + 2.63·16-s − 7.85·17-s + 7.13·19-s + 3.92·20-s + 1.13·22-s + 6.70·23-s − 0.0401·25-s + 2.42·26-s + 1.65·28-s − 2.92·29-s + 3.83·31-s − 4.94·32-s + 3.82·34-s + 2.08·35-s + 5.27·37-s − 3.47·38-s − 4.08·40-s − 7.47·41-s + ⋯ |
| L(s) = 1 | − 0.344·2-s − 0.881·4-s − 0.995·5-s − 0.354·7-s + 0.648·8-s + 0.343·10-s − 0.701·11-s − 1.38·13-s + 0.122·14-s + 0.658·16-s − 1.90·17-s + 1.63·19-s + 0.877·20-s + 0.241·22-s + 1.39·23-s − 0.00802·25-s + 0.475·26-s + 0.312·28-s − 0.543·29-s + 0.689·31-s − 0.874·32-s + 0.655·34-s + 0.352·35-s + 0.867·37-s − 0.564·38-s − 0.645·40-s − 1.16·41-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
| \(L(1)\) |
\(\approx\) |
\(0.5210241910\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5210241910\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.487T + 2T^{2} \) |
| 5 | \( 1 + 2.22T + 5T^{2} \) |
| 7 | \( 1 + 0.937T + 7T^{2} \) |
| 11 | \( 1 + 2.32T + 11T^{2} \) |
| 13 | \( 1 + 4.97T + 13T^{2} \) |
| 17 | \( 1 + 7.85T + 17T^{2} \) |
| 19 | \( 1 - 7.13T + 19T^{2} \) |
| 23 | \( 1 - 6.70T + 23T^{2} \) |
| 29 | \( 1 + 2.92T + 29T^{2} \) |
| 31 | \( 1 - 3.83T + 31T^{2} \) |
| 37 | \( 1 - 5.27T + 37T^{2} \) |
| 41 | \( 1 + 7.47T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 - 6.09T + 47T^{2} \) |
| 53 | \( 1 + 5.27T + 53T^{2} \) |
| 59 | \( 1 + 0.126T + 59T^{2} \) |
| 61 | \( 1 - 0.511T + 61T^{2} \) |
| 67 | \( 1 - 6.08T + 67T^{2} \) |
| 71 | \( 1 - 9.50T + 71T^{2} \) |
| 73 | \( 1 - 15.9T + 73T^{2} \) |
| 79 | \( 1 - 0.569T + 79T^{2} \) |
| 83 | \( 1 - 8.63T + 83T^{2} \) |
| 89 | \( 1 - 16.6T + 89T^{2} \) |
| 97 | \( 1 - 16.3T + 97T^{2} \) |
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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
−9.485845320595649822444918337633, −9.224740645052319224015520792199, −8.044797047408258801856932531679, −7.57768968414707995071054535539, −6.73135714240847355074513870622, −5.20298338108973202811541127135, −4.71090539855963539349398597493, −3.68397449319927052646476477486, −2.55545279869589955129937400283, −0.55572838884657648150857568364,
0.55572838884657648150857568364, 2.55545279869589955129937400283, 3.68397449319927052646476477486, 4.71090539855963539349398597493, 5.20298338108973202811541127135, 6.73135714240847355074513870622, 7.57768968414707995071054535539, 8.044797047408258801856932531679, 9.224740645052319224015520792199, 9.485845320595649822444918337633
