| L(s) = 1 | + 8.16·2-s + 11.0·3-s + 34.6·4-s − 25·5-s + 90.5·6-s + 70.9·7-s + 21.6·8-s − 119.·9-s − 204.·10-s − 121·11-s + 384.·12-s + 1.01e3·13-s + 579.·14-s − 277.·15-s − 931.·16-s − 1.69e3·17-s − 978.·18-s − 361·19-s − 866.·20-s + 787.·21-s − 987.·22-s − 127.·23-s + 240.·24-s + 625·25-s + 8.29e3·26-s − 4.02e3·27-s + 2.45e3·28-s + ⋯ |
| L(s) = 1 | + 1.44·2-s + 0.711·3-s + 1.08·4-s − 0.447·5-s + 1.02·6-s + 0.547·7-s + 0.119·8-s − 0.493·9-s − 0.645·10-s − 0.301·11-s + 0.770·12-s + 1.66·13-s + 0.789·14-s − 0.318·15-s − 0.910·16-s − 1.41·17-s − 0.712·18-s − 0.229·19-s − 0.484·20-s + 0.389·21-s − 0.435·22-s − 0.0502·23-s + 0.0852·24-s + 0.200·25-s + 2.40·26-s − 1.06·27-s + 0.592·28-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
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 8.16T + 32T^{2} \) |
| 3 | \( 1 - 11.0T + 243T^{2} \) |
| 7 | \( 1 - 70.9T + 1.68e4T^{2} \) |
| 13 | \( 1 - 1.01e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.69e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 127.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.31e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 985.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.39e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.12e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 4.66e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.57e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.38e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.64e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.98e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.23e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.33e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.86e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.19e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.07e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.25e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.97e4T + 8.58e9T^{2} \) |
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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
−8.419425788719215779025007538906, −8.231467362077805254362293111860, −6.75722106387398158280854890311, −6.15273090837560695583980378222, −5.08491272454907862316109230729, −4.31883343996599204363988676874, −3.50120883932096584851192942759, −2.76866212394957705687595545298, −1.68283345351229307308605213862, 0,
1.68283345351229307308605213862, 2.76866212394957705687595545298, 3.50120883932096584851192942759, 4.31883343996599204363988676874, 5.08491272454907862316109230729, 6.15273090837560695583980378222, 6.75722106387398158280854890311, 8.231467362077805254362293111860, 8.419425788719215779025007538906
