| L(s) = 1 | + 3.84·2-s − 113.·4-s − 125·5-s + 1.35e3·7-s − 927.·8-s − 480.·10-s − 678.·11-s − 1.09e3·13-s + 5.19e3·14-s + 1.09e4·16-s + 1.86e4·17-s − 2.92e4·19-s + 1.41e4·20-s − 2.60e3·22-s − 4.28e4·23-s + 1.56e4·25-s − 4.20e3·26-s − 1.53e5·28-s + 1.44e5·29-s − 9.27e4·31-s + 1.60e5·32-s + 7.17e4·34-s − 1.69e5·35-s − 3.34e5·37-s − 1.12e5·38-s + 1.15e5·40-s − 4.57e4·41-s + ⋯ |
| L(s) = 1 | + 0.339·2-s − 0.884·4-s − 0.447·5-s + 1.49·7-s − 0.640·8-s − 0.151·10-s − 0.153·11-s − 0.138·13-s + 0.506·14-s + 0.667·16-s + 0.921·17-s − 0.979·19-s + 0.395·20-s − 0.0521·22-s − 0.735·23-s + 0.199·25-s − 0.0469·26-s − 1.31·28-s + 1.10·29-s − 0.559·31-s + 0.866·32-s + 0.312·34-s − 0.666·35-s − 1.08·37-s − 0.332·38-s + 0.286·40-s − 0.103·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.910615212\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.910615212\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 125T \) |
| good | 2 | \( 1 - 3.84T + 128T^{2} \) |
| 7 | \( 1 - 1.35e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 678.T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.09e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.86e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.92e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 4.28e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.44e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 9.27e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.34e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 4.57e4T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.95e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 3.93e4T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.16e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 3.06e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.95e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 5.35e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.24e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.51e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.41e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.82e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 7.18e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.46e6T + 8.07e13T^{2} \) |
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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
−10.17362803902085595527396844392, −8.966994949415162477354137033183, −8.218153507289170677202793443311, −7.59212920823767830414604014014, −6.07047004640485738642854444153, −5.00763307335467481906756797969, −4.43483014788886752220789291515, −3.36513270583666573256892534600, −1.85459321377459191810347151420, −0.61732072831931706956425986805,
0.61732072831931706956425986805, 1.85459321377459191810347151420, 3.36513270583666573256892534600, 4.43483014788886752220789291515, 5.00763307335467481906756797969, 6.07047004640485738642854444153, 7.59212920823767830414604014014, 8.218153507289170677202793443311, 8.966994949415162477354137033183, 10.17362803902085595527396844392
