| L(s) = 1 | + 3.09·2-s + 1.56·4-s − 0.265·5-s + 7·7-s − 19.9·8-s − 0.820·10-s + 11·11-s + 9.99·13-s + 21.6·14-s − 74.0·16-s + 103.·17-s + 54.7·19-s − 0.415·20-s + 34.0·22-s − 26.6·23-s − 124.·25-s + 30.9·26-s + 10.9·28-s − 17.2·29-s + 202.·31-s − 69.8·32-s + 320.·34-s − 1.85·35-s + 244.·37-s + 169.·38-s + 5.28·40-s + 306.·41-s + ⋯ |
| L(s) = 1 | + 1.09·2-s + 0.195·4-s − 0.0237·5-s + 0.377·7-s − 0.879·8-s − 0.0259·10-s + 0.301·11-s + 0.213·13-s + 0.413·14-s − 1.15·16-s + 1.47·17-s + 0.660·19-s − 0.00464·20-s + 0.329·22-s − 0.241·23-s − 0.999·25-s + 0.233·26-s + 0.0739·28-s − 0.110·29-s + 1.17·31-s − 0.385·32-s + 1.61·34-s − 0.00897·35-s + 1.08·37-s + 0.722·38-s + 0.0208·40-s + 1.16·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.495316022\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.495316022\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 - 3.09T + 8T^{2} \) |
| 5 | \( 1 + 0.265T + 125T^{2} \) |
| 13 | \( 1 - 9.99T + 2.19e3T^{2} \) |
| 17 | \( 1 - 103.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 54.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 26.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 17.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 202.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 244.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 306.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 330.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 74.6T + 1.03e5T^{2} \) |
| 53 | \( 1 - 428.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 350.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 153.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 192.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 821.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 727.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 410.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 289.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 225.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 420.T + 9.12e5T^{2} \) |
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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
−10.00839544703775419088322859019, −9.314717284416127258382817733924, −8.217696754652972852149057034554, −7.39764088670882584389900141373, −6.07965505672977283676181089883, −5.56302377624991190886534377699, −4.47366350300789552851419321658, −3.69109725887804620821313044823, −2.61092905111339169472971664367, −0.960717694687656106476898507657,
0.960717694687656106476898507657, 2.61092905111339169472971664367, 3.69109725887804620821313044823, 4.47366350300789552851419321658, 5.56302377624991190886534377699, 6.07965505672977283676181089883, 7.39764088670882584389900141373, 8.217696754652972852149057034554, 9.314717284416127258382817733924, 10.00839544703775419088322859019
