| L(s) = 1 | − 1.53·2-s + 81·3-s − 509.·4-s + 521.·5-s − 124.·6-s + 1.13e4·7-s + 1.56e3·8-s + 6.56e3·9-s − 800.·10-s + 9.51e4·11-s − 4.12e4·12-s + 5.89e4·13-s − 1.74e4·14-s + 4.22e4·15-s + 2.58e5·16-s − 8.92e4·17-s − 1.00e4·18-s + 3.08e5·19-s − 2.65e5·20-s + 9.19e5·21-s − 1.46e5·22-s + 2.23e6·23-s + 1.27e5·24-s − 1.68e6·25-s − 9.04e4·26-s + 5.31e5·27-s − 5.78e6·28-s + ⋯ |
| L(s) = 1 | − 0.0678·2-s + 0.577·3-s − 0.995·4-s + 0.373·5-s − 0.0391·6-s + 1.78·7-s + 0.135·8-s + 0.333·9-s − 0.0253·10-s + 1.95·11-s − 0.574·12-s + 0.572·13-s − 0.121·14-s + 0.215·15-s + 0.986·16-s − 0.259·17-s − 0.0226·18-s + 0.543·19-s − 0.371·20-s + 1.03·21-s − 0.132·22-s + 1.66·23-s + 0.0781·24-s − 0.860·25-s − 0.0388·26-s + 0.192·27-s − 1.77·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(3.635091070\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.635091070\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 81T \) |
| 59 | \( 1 + 1.21e7T \) |
| good | 2 | \( 1 + 1.53T + 512T^{2} \) |
| 5 | \( 1 - 521.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 1.13e4T + 4.03e7T^{2} \) |
| 11 | \( 1 - 9.51e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 5.89e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 8.92e4T + 1.18e11T^{2} \) |
| 19 | \( 1 - 3.08e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.23e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 7.46e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 6.83e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.22e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 5.99e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.75e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.84e6T + 1.11e15T^{2} \) |
| 53 | \( 1 - 2.55e7T + 3.29e15T^{2} \) |
| 61 | \( 1 - 7.45e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 7.95e5T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.30e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 5.28e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.07e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 6.59e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 5.89e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.40e9T + 7.60e17T^{2} \) |
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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
−11.10401839714127661033357752229, −9.688811202943410782771380625916, −8.911914015617596895023751991555, −8.292462626955461413653695809205, −7.07825514115617031436021520560, −5.52485681505150781876834174626, −4.48730663816803498285077990789, −3.61799416615980755237759769659, −1.70340853377238033158432057115, −1.08958825660705543096588106000,
1.08958825660705543096588106000, 1.70340853377238033158432057115, 3.61799416615980755237759769659, 4.48730663816803498285077990789, 5.52485681505150781876834174626, 7.07825514115617031436021520560, 8.292462626955461413653695809205, 8.911914015617596895023751991555, 9.688811202943410782771380625916, 11.10401839714127661033357752229
