| L(s) = 1 | − 0.259·2-s + 8.53·3-s − 31.9·4-s − 25·5-s − 2.21·6-s − 211.·7-s + 16.5·8-s − 170.·9-s + 6.47·10-s + 121·11-s − 272.·12-s + 462.·13-s + 54.9·14-s − 213.·15-s + 1.01e3·16-s + 991.·17-s + 44.0·18-s + 361·19-s + 798.·20-s − 1.80e3·21-s − 31.3·22-s − 1.76e3·23-s + 141.·24-s + 625·25-s − 119.·26-s − 3.52e3·27-s + 6.76e3·28-s + ⋯ |
| L(s) = 1 | − 0.0457·2-s + 0.547·3-s − 0.997·4-s − 0.447·5-s − 0.0250·6-s − 1.63·7-s + 0.0914·8-s − 0.700·9-s + 0.0204·10-s + 0.301·11-s − 0.546·12-s + 0.759·13-s + 0.0748·14-s − 0.244·15-s + 0.993·16-s + 0.831·17-s + 0.0320·18-s + 0.229·19-s + 0.446·20-s − 0.895·21-s − 0.0138·22-s − 0.695·23-s + 0.0501·24-s + 0.200·25-s − 0.0347·26-s − 0.931·27-s + 1.63·28-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 | \( 1 + 0.259T + 32T^{2} \) |
| 3 | \( 1 - 8.53T + 243T^{2} \) |
| 7 | \( 1 + 211.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 462.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 991.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 1.76e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.28e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.71e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.73e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 3.84e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.72e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 7.53e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.72e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.24e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.20e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.67e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.99e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.16e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.48e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.78e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.69e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.32e3T + 8.58e9T^{2} \) |
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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
−8.827027238500536228777576507089, −8.160271858335057092738057237501, −7.24099499825211691183915505898, −6.08001363431949317877549198424, −5.48672493744815767609734339021, −3.88714318735535994827930391897, −3.65712536748687170892462863369, −2.64161594539060370404799014516, −0.909449271463736154565293003626, 0,
0.909449271463736154565293003626, 2.64161594539060370404799014516, 3.65712536748687170892462863369, 3.88714318735535994827930391897, 5.48672493744815767609734339021, 6.08001363431949317877549198424, 7.24099499825211691183915505898, 8.160271858335057092738057237501, 8.827027238500536228777576507089
