| L(s) = 1 | − 7.82·2-s + 29.2·4-s + 25·5-s − 125.·7-s + 21.7·8-s − 195.·10-s + 121·11-s + 532.·13-s + 981.·14-s − 1.10e3·16-s + 1.37e3·17-s − 554.·19-s + 730.·20-s − 946.·22-s − 4.25e3·23-s + 625·25-s − 4.16e3·26-s − 3.66e3·28-s + 6.97e3·29-s + 3.13e3·31-s + 7.95e3·32-s − 1.07e4·34-s − 3.13e3·35-s + 1.38e3·37-s + 4.33e3·38-s + 542.·40-s − 679.·41-s + ⋯ |
| L(s) = 1 | − 1.38·2-s + 0.913·4-s + 0.447·5-s − 0.967·7-s + 0.119·8-s − 0.618·10-s + 0.301·11-s + 0.873·13-s + 1.33·14-s − 1.07·16-s + 1.15·17-s − 0.352·19-s + 0.408·20-s − 0.417·22-s − 1.67·23-s + 0.200·25-s − 1.20·26-s − 0.883·28-s + 1.53·29-s + 0.584·31-s + 1.37·32-s − 1.59·34-s − 0.432·35-s + 0.166·37-s + 0.487·38-s + 0.0536·40-s − 0.0631·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.9267862478\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9267862478\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| 11 | \( 1 - 121T \) |
| good | 2 | \( 1 + 7.82T + 32T^{2} \) |
| 7 | \( 1 + 125.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 532.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.37e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 554.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.25e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.97e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.13e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.38e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 679.T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.72e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.51e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 9.54e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.75e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.05e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.87e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.25e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.37e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.86e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.22e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.15e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.90e4T + 8.58e9T^{2} \) |
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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
−10.11627495204726796977795062628, −9.308343577626767925884025247419, −8.506620099248683383634515387897, −7.70729665574116560064532132612, −6.55323250359811033740545493954, −5.92919040361224016990541103376, −4.29891333718899468082564518790, −3.00236351208165432035439806510, −1.64698842712430161010260958272, −0.62372095821309887457002522925,
0.62372095821309887457002522925, 1.64698842712430161010260958272, 3.00236351208165432035439806510, 4.29891333718899468082564518790, 5.92919040361224016990541103376, 6.55323250359811033740545493954, 7.70729665574116560064532132612, 8.506620099248683383634515387897, 9.308343577626767925884025247419, 10.11627495204726796977795062628
