| L(s) = 1 | + (15.0 + 5.47i)2-s + (5.37 + 30.4i)3-s + (196. + 164. i)4-s + (−860. + 722. i)5-s + (−85.9 + 487. i)6-s + (6.05e3 − 1.04e4i)7-s + (2.04e3 + 3.54e3i)8-s + (1.75e4 − 6.40e3i)9-s + (−1.68e4 + 6.14e3i)10-s + (2.84e4 + 4.92e4i)11-s + (−3.96e3 + 6.85e3i)12-s + (−2.80e4 + 1.59e5i)13-s + (1.48e5 − 1.24e5i)14-s + (−2.66e4 − 2.23e4i)15-s + (1.13e4 + 6.45e4i)16-s + (−1.24e5 − 4.54e4i)17-s + ⋯ |
| L(s) = 1 | + (0.664 + 0.241i)2-s + (0.0382 + 0.217i)3-s + (0.383 + 0.321i)4-s + (−0.615 + 0.516i)5-s + (−0.0270 + 0.153i)6-s + (0.953 − 1.65i)7-s + (0.176 + 0.306i)8-s + (0.893 − 0.325i)9-s + (−0.534 + 0.194i)10-s + (0.585 + 1.01i)11-s + (−0.0551 + 0.0954i)12-s + (−0.272 + 1.54i)13-s + (1.03 − 0.866i)14-s + (−0.135 − 0.113i)15-s + (0.0434 + 0.246i)16-s + (−0.362 − 0.131i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.629 - 0.777i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.629 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.79608 + 1.33338i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.79608 + 1.33338i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-15.0 - 5.47i)T \) |
| 19 | \( 1 + (-5.43e5 - 1.63e5i)T \) |
| good | 3 | \( 1 + (-5.37 - 30.4i)T + (-1.84e4 + 6.73e3i)T^{2} \) |
| 5 | \( 1 + (860. - 722. i)T + (3.39e5 - 1.92e6i)T^{2} \) |
| 7 | \( 1 + (-6.05e3 + 1.04e4i)T + (-2.01e7 - 3.49e7i)T^{2} \) |
| 11 | \( 1 + (-2.84e4 - 4.92e4i)T + (-1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 + (2.80e4 - 1.59e5i)T + (-9.96e9 - 3.62e9i)T^{2} \) |
| 17 | \( 1 + (1.24e5 + 4.54e4i)T + (9.08e10 + 7.62e10i)T^{2} \) |
| 23 | \( 1 + (-5.90e5 - 4.95e5i)T + (3.12e11 + 1.77e12i)T^{2} \) |
| 29 | \( 1 + (9.80e5 - 3.57e5i)T + (1.11e13 - 9.32e12i)T^{2} \) |
| 31 | \( 1 + (-3.39e6 + 5.87e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + 3.62e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + (-2.99e6 - 1.70e7i)T + (-3.07e14 + 1.11e14i)T^{2} \) |
| 43 | \( 1 + (-2.81e7 + 2.36e7i)T + (8.72e13 - 4.94e14i)T^{2} \) |
| 47 | \( 1 + (-4.46e7 + 1.62e7i)T + (8.57e14 - 7.19e14i)T^{2} \) |
| 53 | \( 1 + (2.16e7 + 1.81e7i)T + (5.72e14 + 3.24e15i)T^{2} \) |
| 59 | \( 1 + (1.41e8 + 5.14e7i)T + (6.63e15 + 5.56e15i)T^{2} \) |
| 61 | \( 1 + (-4.67e7 - 3.92e7i)T + (2.03e15 + 1.15e16i)T^{2} \) |
| 67 | \( 1 + (7.66e7 - 2.78e7i)T + (2.08e16 - 1.74e16i)T^{2} \) |
| 71 | \( 1 + (2.23e8 - 1.87e8i)T + (7.96e15 - 4.51e16i)T^{2} \) |
| 73 | \( 1 + (-1.24e6 - 7.04e6i)T + (-5.53e16 + 2.01e16i)T^{2} \) |
| 79 | \( 1 + (8.73e7 + 4.95e8i)T + (-1.12e17 + 4.09e16i)T^{2} \) |
| 83 | \( 1 + (2.68e8 - 4.65e8i)T + (-9.34e16 - 1.61e17i)T^{2} \) |
| 89 | \( 1 + (9.29e7 - 5.26e8i)T + (-3.29e17 - 1.19e17i)T^{2} \) |
| 97 | \( 1 + (1.33e9 + 4.85e8i)T + (5.82e17 + 4.88e17i)T^{2} \) |
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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
−14.45749183731678271957217716364, −13.59063057644339804425732104448, −11.96541658288345960330725217017, −11.04582573162657808156811006259, −9.645613178125687677357578678868, −7.37710284022653635114560370762, −7.05810230930703369819734377684, −4.48175036484071643717008087824, −3.95678282800006116994683068419, −1.48818374629632982251807197107,
1.10930284632934373936907536077, 2.80438415618035430049340937523, 4.68238844403807907164165555932, 5.78308090110642442888998500914, 7.78374210627250196571549477194, 8.894449845499317074372355695532, 10.82767948750664637450783780454, 12.01492425744201324310078309369, 12.65605122067862825597210560185, 14.09414874174384120992992045867
