| L(s) = 1 | + (−2.73 − 0.732i)2-s + (0.661 + 1.14i)3-s + (6.92 + 4i)4-s + (18.3 − 18.3i)5-s + (−0.968 − 3.61i)6-s + (57.6 − 15.4i)7-s + (−15.9 − 16i)8-s + (39.6 − 68.6i)9-s + (−63.6 + 36.7i)10-s + (−15.9 + 59.4i)11-s + 10.5i·12-s + (−46.0 + 162. i)13-s − 168.·14-s + (33.2 + 8.89i)15-s + (31.9 + 55.4i)16-s + (−276. − 159. i)17-s + ⋯ |
| L(s) = 1 | + (−0.683 − 0.183i)2-s + (0.0735 + 0.127i)3-s + (0.433 + 0.250i)4-s + (0.734 − 0.734i)5-s + (−0.0269 − 0.100i)6-s + (1.17 − 0.315i)7-s + (−0.249 − 0.250i)8-s + (0.489 − 0.847i)9-s + (−0.636 + 0.367i)10-s + (−0.131 + 0.490i)11-s + 0.0735i·12-s + (−0.272 + 0.962i)13-s − 0.860·14-s + (0.147 + 0.0395i)15-s + (0.124 + 0.216i)16-s + (−0.957 − 0.552i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 + 0.534i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.845 + 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.15522 - 0.334271i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.15522 - 0.334271i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.73 + 0.732i)T \) |
| 13 | \( 1 + (46.0 - 162. i)T \) |
| good | 3 | \( 1 + (-0.661 - 1.14i)T + (-40.5 + 70.1i)T^{2} \) |
| 5 | \( 1 + (-18.3 + 18.3i)T - 625iT^{2} \) |
| 7 | \( 1 + (-57.6 + 15.4i)T + (2.07e3 - 1.20e3i)T^{2} \) |
| 11 | \( 1 + (15.9 - 59.4i)T + (-1.26e4 - 7.32e3i)T^{2} \) |
| 17 | \( 1 + (276. + 159. i)T + (4.17e4 + 7.23e4i)T^{2} \) |
| 19 | \( 1 + (-21.4 - 80.1i)T + (-1.12e5 + 6.51e4i)T^{2} \) |
| 23 | \( 1 + (-378. + 218. i)T + (1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 + (194. + 336. i)T + (-3.53e5 + 6.12e5i)T^{2} \) |
| 31 | \( 1 + (771. - 771. i)T - 9.23e5iT^{2} \) |
| 37 | \( 1 + (508. - 1.89e3i)T + (-1.62e6 - 9.37e5i)T^{2} \) |
| 41 | \( 1 + (2.77e3 + 743. i)T + (2.44e6 + 1.41e6i)T^{2} \) |
| 43 | \( 1 + (-850. - 490. i)T + (1.70e6 + 2.96e6i)T^{2} \) |
| 47 | \( 1 + (-3.07e3 - 3.07e3i)T + 4.87e6iT^{2} \) |
| 53 | \( 1 + 3.41e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + (-1.93e3 + 519. i)T + (1.04e7 - 6.05e6i)T^{2} \) |
| 61 | \( 1 + (-2.46e3 + 4.26e3i)T + (-6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-3.34e3 - 896. i)T + (1.74e7 + 1.00e7i)T^{2} \) |
| 71 | \( 1 + (1.16e3 + 4.33e3i)T + (-2.20e7 + 1.27e7i)T^{2} \) |
| 73 | \( 1 + (-2.88e3 - 2.88e3i)T + 2.83e7iT^{2} \) |
| 79 | \( 1 + 1.12e4T + 3.89e7T^{2} \) |
| 83 | \( 1 + (5.38e3 - 5.38e3i)T - 4.74e7iT^{2} \) |
| 89 | \( 1 + (382. - 1.42e3i)T + (-5.43e7 - 3.13e7i)T^{2} \) |
| 97 | \( 1 + (553. + 2.06e3i)T + (-7.66e7 + 4.42e7i)T^{2} \) |
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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
−16.98464650389645304347016752644, −15.49185090061274433969338619868, −14.12634338571066645116842907645, −12.62103713706792537735193691616, −11.28232435078502351578359377090, −9.734526205434438337528428058019, −8.748147625595644260450675320617, −6.98480605518299209022907467918, −4.70956776269003809511806438930, −1.58410679422635846404602421290,
2.12014543252443693475666799095, 5.45838143380368229655678063840, 7.28429614224748125529898161939, 8.596069726206489275565398643500, 10.35051910104421037064366336556, 11.16903773900078320088768525873, 13.19053490912759005760547500665, 14.49198293449455782309202988156, 15.54141524632373166492028781044, 17.14755869651025804418888301470
