| 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
−17.14755869651025804418888301470, −15.54141524632373166492028781044, −14.49198293449455782309202988156, −13.19053490912759005760547500665, −11.16903773900078320088768525873, −10.35051910104421037064366336556, −8.596069726206489275565398643500, −7.28429614224748125529898161939, −5.45838143380368229655678063840, −2.12014543252443693475666799095,
1.58410679422635846404602421290, 4.70956776269003809511806438930, 6.98480605518299209022907467918, 8.748147625595644260450675320617, 9.734526205434438337528428058019, 11.28232435078502351578359377090, 12.62103713706792537735193691616, 14.12634338571066645116842907645, 15.49185090061274433969338619868, 16.98464650389645304347016752644
