| L(s) = 1 | + (1.65 − 1.20i)2-s + (−5.47 + 7.53i)3-s + (−3.64 + 11.2i)4-s − 37.3·5-s + 19.0i·6-s + (8.17 − 25.1i)7-s + (17.6 + 54.1i)8-s + (−1.76 − 5.41i)9-s + (−61.9 + 45.0i)10-s + (153. + 49.9i)11-s + (−64.5 − 88.9i)12-s + (−42.3 + 58.2i)13-s + (−16.7 − 51.5i)14-s + (204. − 281. i)15-s + (−58.3 − 42.3i)16-s + (−41.1 + 13.3i)17-s + ⋯ |
| L(s) = 1 | + (0.414 − 0.301i)2-s + (−0.608 + 0.836i)3-s + (−0.227 + 0.701i)4-s − 1.49·5-s + 0.529i·6-s + (0.166 − 0.513i)7-s + (0.275 + 0.846i)8-s + (−0.0217 − 0.0668i)9-s + (−0.619 + 0.450i)10-s + (1.26 + 0.412i)11-s + (−0.448 − 0.617i)12-s + (−0.250 + 0.344i)13-s + (−0.0854 − 0.263i)14-s + (0.909 − 1.25i)15-s + (−0.227 − 0.165i)16-s + (−0.142 + 0.0462i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.560 - 0.828i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.560 - 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.408482 + 0.769913i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.408482 + 0.769913i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 + (-245. - 929. i)T \) |
| good | 2 | \( 1 + (-1.65 + 1.20i)T + (4.94 - 15.2i)T^{2} \) |
| 3 | \( 1 + (5.47 - 7.53i)T + (-25.0 - 77.0i)T^{2} \) |
| 5 | \( 1 + 37.3T + 625T^{2} \) |
| 7 | \( 1 + (-8.17 + 25.1i)T + (-1.94e3 - 1.41e3i)T^{2} \) |
| 11 | \( 1 + (-153. - 49.9i)T + (1.18e4 + 8.60e3i)T^{2} \) |
| 13 | \( 1 + (42.3 - 58.2i)T + (-8.82e3 - 2.71e4i)T^{2} \) |
| 17 | \( 1 + (41.1 - 13.3i)T + (6.75e4 - 4.90e4i)T^{2} \) |
| 19 | \( 1 + (385. - 280. i)T + (4.02e4 - 1.23e5i)T^{2} \) |
| 23 | \( 1 + (-440. + 143. i)T + (2.26e5 - 1.64e5i)T^{2} \) |
| 29 | \( 1 + (-176. - 242. i)T + (-2.18e5 + 6.72e5i)T^{2} \) |
| 37 | \( 1 - 1.52e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (-2.16e3 + 1.57e3i)T + (8.73e5 - 2.68e6i)T^{2} \) |
| 43 | \( 1 + (1.49e3 + 2.05e3i)T + (-1.05e6 + 3.25e6i)T^{2} \) |
| 47 | \( 1 + (-2.52e3 - 1.83e3i)T + (1.50e6 + 4.64e6i)T^{2} \) |
| 53 | \( 1 + (-3.79e3 + 1.23e3i)T + (6.38e6 - 4.63e6i)T^{2} \) |
| 59 | \( 1 + (3.81e3 + 2.77e3i)T + (3.74e6 + 1.15e7i)T^{2} \) |
| 61 | \( 1 - 1.14e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 1.36e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + (-1.10e3 - 3.39e3i)T + (-2.05e7 + 1.49e7i)T^{2} \) |
| 73 | \( 1 + (2.66e3 + 866. i)T + (2.29e7 + 1.66e7i)T^{2} \) |
| 79 | \( 1 + (6.81e3 - 2.21e3i)T + (3.15e7 - 2.28e7i)T^{2} \) |
| 83 | \( 1 + (-4.79e3 - 6.60e3i)T + (-1.46e7 + 4.51e7i)T^{2} \) |
| 89 | \( 1 + (9.59e3 + 3.11e3i)T + (5.07e7 + 3.68e7i)T^{2} \) |
| 97 | \( 1 + (-3.67e3 + 1.13e4i)T + (-7.16e7 - 5.20e7i)T^{2} \) |
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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
−16.69673716974019320965071779370, −15.45186896937068138472673192128, −14.22875449331145426561224667101, −12.46038082177012106223562833643, −11.65181365081586256867002839376, −10.64719699517913879623739575083, −8.679723456963771172892324972205, −7.23063831831867706632280411635, −4.54516438770748413929603119015, −3.89956309811085086973703218158,
0.64636954092237250017662352884, 4.25145975683844321349513872777, 6.07568816459451599381230819425, 7.25079263966006153168861920477, 8.990161132192119983367115345968, 11.13342567168598884729395004319, 12.00205269234611013317414414762, 13.14326888918410284380293582987, 14.80403663682786395353165939097, 15.39958185929440769642937454910
