| L(s) = 1 | − 33.5·2-s + 610.·4-s − 625·5-s + 2.40e3·7-s − 3.29e3·8-s + 2.09e4·10-s + 3.65e4·11-s + 1.21e4·13-s − 8.04e4·14-s − 2.02e5·16-s + 1.51e5·17-s + 5.18e5·19-s − 3.81e5·20-s − 1.22e6·22-s + 2.13e6·23-s + 3.90e5·25-s − 4.05e5·26-s + 1.46e6·28-s + 1.31e6·29-s + 3.38e6·31-s + 8.45e6·32-s − 5.07e6·34-s − 1.50e6·35-s − 1.53e7·37-s − 1.73e7·38-s + 2.06e6·40-s + 2.82e7·41-s + ⋯ |
| L(s) = 1 | − 1.48·2-s + 1.19·4-s − 0.447·5-s + 0.377·7-s − 0.284·8-s + 0.662·10-s + 0.752·11-s + 0.117·13-s − 0.559·14-s − 0.770·16-s + 0.439·17-s + 0.913·19-s − 0.533·20-s − 1.11·22-s + 1.58·23-s + 0.200·25-s − 0.174·26-s + 0.450·28-s + 0.344·29-s + 0.658·31-s + 1.42·32-s − 0.651·34-s − 0.169·35-s − 1.34·37-s − 1.35·38-s + 0.127·40-s + 1.56·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.275729707\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.275729707\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 625T \) |
| 7 | \( 1 - 2.40e3T \) |
| good | 2 | \( 1 + 33.5T + 512T^{2} \) |
| 11 | \( 1 - 3.65e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.21e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 1.51e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 5.18e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.13e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 1.31e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 3.38e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.53e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.82e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.53e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 7.18e6T + 1.11e15T^{2} \) |
| 53 | \( 1 - 5.45e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 9.04e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 8.16e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.97e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.51e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.43e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.93e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 1.09e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 7.26e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 4.86e7T + 7.60e17T^{2} \) |
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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
−9.912514027893697623035403283956, −9.088180301230093558217693700613, −8.388085550213483326800043100223, −7.44840059384876898286894215242, −6.77554876911984755284774737700, −5.28864974654885368183646658505, −4.04216900223873078446460560417, −2.69567447403248851525737045114, −1.28848320419624872264300591985, −0.73429334360272286728693362583,
0.73429334360272286728693362583, 1.28848320419624872264300591985, 2.69567447403248851525737045114, 4.04216900223873078446460560417, 5.28864974654885368183646658505, 6.77554876911984755284774737700, 7.44840059384876898286894215242, 8.388085550213483326800043100223, 9.088180301230093558217693700613, 9.912514027893697623035403283956
