| L(s) = 1 | − 8·2-s − 40.7·3-s + 64·4-s − 417.·5-s + 326.·6-s − 620.·7-s − 512·8-s − 522.·9-s + 3.34e3·10-s − 7.91e3·11-s − 2.61e3·12-s − 1.38e3·13-s + 4.96e3·14-s + 1.70e4·15-s + 4.09e3·16-s − 3.46e4·17-s + 4.18e3·18-s − 6.52e3·19-s − 2.67e4·20-s + 2.53e4·21-s + 6.33e4·22-s − 9.79e3·23-s + 2.08e4·24-s + 9.64e4·25-s + 1.11e4·26-s + 1.10e5·27-s − 3.97e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.872·3-s + 0.5·4-s − 1.49·5-s + 0.616·6-s − 0.684·7-s − 0.353·8-s − 0.238·9-s + 1.05·10-s − 1.79·11-s − 0.436·12-s − 0.175·13-s + 0.483·14-s + 1.30·15-s + 0.250·16-s − 1.70·17-s + 0.168·18-s − 0.218·19-s − 0.747·20-s + 0.596·21-s + 1.26·22-s − 0.167·23-s + 0.308·24-s + 1.23·25-s + 0.123·26-s + 1.08·27-s − 0.342·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.002382153068\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.002382153068\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8T \) |
| 31 | \( 1 + 2.97e4T \) |
| good | 3 | \( 1 + 40.7T + 2.18e3T^{2} \) |
| 5 | \( 1 + 417.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 620.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 7.91e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.38e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.46e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 6.52e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 9.79e3T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.03e5T + 1.72e10T^{2} \) |
| 37 | \( 1 - 2.99e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 5.25e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.85e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 2.28e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 7.19e3T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.73e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.62e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.22e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.06e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.92e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 5.93e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 9.72e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 5.66e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.33e7T + 8.07e13T^{2} \) |
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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
−13.18402797660072253186475829567, −12.09282825178758933246158724972, −11.16013396577547618969800939173, −10.39279142761971456308733789961, −8.662343314373201026788298251632, −7.63277654859942154501364671890, −6.38566289369355687387499095341, −4.74416666258073524725279468891, −2.86864830486608979768701266036, −0.03600905635187031383214116715,
0.03600905635187031383214116715, 2.86864830486608979768701266036, 4.74416666258073524725279468891, 6.38566289369355687387499095341, 7.63277654859942154501364671890, 8.662343314373201026788298251632, 10.39279142761971456308733789961, 11.16013396577547618969800939173, 12.09282825178758933246158724972, 13.18402797660072253186475829567
