| L(s) = 1 | + 5.27·2-s + 0.134·3-s − 4.18·4-s − 25·5-s + 0.711·6-s − 143.·7-s − 190.·8-s − 242.·9-s − 131.·10-s + 121·11-s − 0.565·12-s − 147.·13-s − 758.·14-s − 3.37·15-s − 872.·16-s − 1.42e3·17-s − 1.28e3·18-s − 361·19-s + 104.·20-s − 19.3·21-s + 638.·22-s − 469.·23-s − 25.7·24-s + 625·25-s − 779.·26-s − 65.5·27-s + 602.·28-s + ⋯ |
| L(s) = 1 | + 0.932·2-s + 0.00865·3-s − 0.130·4-s − 0.447·5-s + 0.00806·6-s − 1.10·7-s − 1.05·8-s − 0.999·9-s − 0.416·10-s + 0.301·11-s − 0.00113·12-s − 0.242·13-s − 1.03·14-s − 0.00387·15-s − 0.852·16-s − 1.19·17-s − 0.932·18-s − 0.229·19-s + 0.0585·20-s − 0.00959·21-s + 0.281·22-s − 0.184·23-s − 0.00912·24-s + 0.200·25-s − 0.226·26-s − 0.0173·27-s + 0.145·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0007087602598\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0007087602598\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 + 361T \) |
| good | 2 | \( 1 - 5.27T + 32T^{2} \) |
| 3 | \( 1 - 0.134T + 243T^{2} \) |
| 7 | \( 1 + 143.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 147.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.42e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 469.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 8.50e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.48e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.51e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.21e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.13e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.07e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.09e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.48e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.29e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.03e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.62e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.13e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 510.T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.77e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.75e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.60e5T + 8.58e9T^{2} \) |
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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
−8.978176820502759579478698424790, −8.671560520069607647835049349890, −7.27802759853561824974158509593, −6.45371661943844476964727489050, −5.70530787655860371938053373334, −4.84640406689824004568442179165, −3.74175657033946357147880835514, −3.30618449647980903992484079471, −2.15134542329527668391316830874, −0.009193166371229907585281786377,
0.009193166371229907585281786377, 2.15134542329527668391316830874, 3.30618449647980903992484079471, 3.74175657033946357147880835514, 4.84640406689824004568442179165, 5.70530787655860371938053373334, 6.45371661943844476964727489050, 7.27802759853561824974158509593, 8.671560520069607647835049349890, 8.978176820502759579478698424790
