| L(s) = 1 | − 31.9·2-s + 275.·3-s + 507.·4-s + 217.·5-s − 8.81e3·6-s + 4.69e3·7-s + 140.·8-s + 5.64e4·9-s − 6.94e3·10-s + 2.78e4·11-s + 1.40e5·12-s − 1.44e4·13-s − 1.50e5·14-s + 6.00e4·15-s − 2.64e5·16-s − 3.22e5·17-s − 1.80e6·18-s + 4.60e5·19-s + 1.10e5·20-s + 1.29e6·21-s − 8.88e5·22-s + 7.67e5·23-s + 3.87e4·24-s − 1.90e6·25-s + 4.60e5·26-s + 1.01e7·27-s + 2.38e6·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.96·3-s + 0.991·4-s + 0.155·5-s − 2.77·6-s + 0.739·7-s + 0.0121·8-s + 2.86·9-s − 0.219·10-s + 0.573·11-s + 1.95·12-s − 0.139·13-s − 1.04·14-s + 0.306·15-s − 1.00·16-s − 0.937·17-s − 4.04·18-s + 0.810·19-s + 0.154·20-s + 1.45·21-s − 0.808·22-s + 0.571·23-s + 0.0238·24-s − 0.975·25-s + 0.197·26-s + 3.67·27-s + 0.733·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.241433440\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.241433440\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 - 3.41e6T \) |
| good | 2 | \( 1 + 31.9T + 512T^{2} \) |
| 3 | \( 1 - 275.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 217.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 4.69e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 2.78e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.44e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 3.22e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 4.60e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 7.67e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 3.26e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 5.37e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.95e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.56e7T + 3.27e14T^{2} \) |
| 47 | \( 1 - 2.57e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.77e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 6.03e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 3.61e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.45e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.60e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.99e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.60e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 4.07e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 3.56e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.68e9T + 7.60e17T^{2} \) |
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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
−14.15289220114845565738097371731, −13.12574910155119156560306783265, −11.10706021795998058112835551305, −9.609953692199949915995331154692, −9.100057413015266892366618042240, −8.011195025561553682663549822536, −7.23482067241577095377829434035, −4.17396890410989625603324603989, −2.35690753419799461394901964682, −1.31454160449396383283479495990,
1.31454160449396383283479495990, 2.35690753419799461394901964682, 4.17396890410989625603324603989, 7.23482067241577095377829434035, 8.011195025561553682663549822536, 9.100057413015266892366618042240, 9.609953692199949915995331154692, 11.10706021795998058112835551305, 13.12574910155119156560306783265, 14.15289220114845565738097371731
