| L(s) = 1 | + (15.0 − 5.47i)2-s + (−29.8 + 169. i)3-s + (196. − 164. i)4-s + (−424. − 355. i)5-s + (477. + 2.71e3i)6-s + (−4.17e3 − 7.23e3i)7-s + (2.04e3 − 3.54e3i)8-s + (−9.31e3 − 3.38e3i)9-s + (−8.32e3 − 3.02e3i)10-s + (1.26e4 − 2.19e4i)11-s + (2.20e4 + 3.81e4i)12-s + (−9.30e3 − 5.27e4i)13-s + (−1.02e5 − 8.58e4i)14-s + (7.29e4 − 6.12e4i)15-s + (1.13e4 − 6.45e4i)16-s + (2.83e5 − 1.03e5i)17-s + ⋯ |
| L(s) = 1 | + (0.664 − 0.241i)2-s + (−0.212 + 1.20i)3-s + (0.383 − 0.321i)4-s + (−0.303 − 0.254i)5-s + (0.150 + 0.853i)6-s + (−0.657 − 1.13i)7-s + (0.176 − 0.306i)8-s + (−0.473 − 0.172i)9-s + (−0.263 − 0.0958i)10-s + (0.260 − 0.451i)11-s + (0.306 + 0.530i)12-s + (−0.0903 − 0.512i)13-s + (−0.712 − 0.597i)14-s + (0.372 − 0.312i)15-s + (0.0434 − 0.246i)16-s + (0.823 − 0.299i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.373 + 0.927i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.373 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.64370 - 1.10992i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.64370 - 1.10992i\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 + (-5.68e5 - 2.89e3i)T \) |
| good | 3 | \( 1 + (29.8 - 169. i)T + (-1.84e4 - 6.73e3i)T^{2} \) |
| 5 | \( 1 + (424. + 355. i)T + (3.39e5 + 1.92e6i)T^{2} \) |
| 7 | \( 1 + (4.17e3 + 7.23e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (-1.26e4 + 2.19e4i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 + (9.30e3 + 5.27e4i)T + (-9.96e9 + 3.62e9i)T^{2} \) |
| 17 | \( 1 + (-2.83e5 + 1.03e5i)T + (9.08e10 - 7.62e10i)T^{2} \) |
| 23 | \( 1 + (-4.73e5 + 3.97e5i)T + (3.12e11 - 1.77e12i)T^{2} \) |
| 29 | \( 1 + (3.92e6 + 1.42e6i)T + (1.11e13 + 9.32e12i)T^{2} \) |
| 31 | \( 1 + (3.68e6 + 6.37e6i)T + (-1.32e13 + 2.28e13i)T^{2} \) |
| 37 | \( 1 - 8.42e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + (-9.60e5 + 5.44e6i)T + (-3.07e14 - 1.11e14i)T^{2} \) |
| 43 | \( 1 + (2.03e7 + 1.70e7i)T + (8.72e13 + 4.94e14i)T^{2} \) |
| 47 | \( 1 + (-2.58e7 - 9.42e6i)T + (8.57e14 + 7.19e14i)T^{2} \) |
| 53 | \( 1 + (5.24e7 - 4.40e7i)T + (5.72e14 - 3.24e15i)T^{2} \) |
| 59 | \( 1 + (-5.09e7 + 1.85e7i)T + (6.63e15 - 5.56e15i)T^{2} \) |
| 61 | \( 1 + (7.36e7 - 6.17e7i)T + (2.03e15 - 1.15e16i)T^{2} \) |
| 67 | \( 1 + (-9.73e6 - 3.54e6i)T + (2.08e16 + 1.74e16i)T^{2} \) |
| 71 | \( 1 + (-1.17e8 - 9.88e7i)T + (7.96e15 + 4.51e16i)T^{2} \) |
| 73 | \( 1 + (4.18e7 - 2.37e8i)T + (-5.53e16 - 2.01e16i)T^{2} \) |
| 79 | \( 1 + (-3.16e7 + 1.79e8i)T + (-1.12e17 - 4.09e16i)T^{2} \) |
| 83 | \( 1 + (2.76e8 + 4.79e8i)T + (-9.34e16 + 1.61e17i)T^{2} \) |
| 89 | \( 1 + (-7.11e7 - 4.03e8i)T + (-3.29e17 + 1.19e17i)T^{2} \) |
| 97 | \( 1 + (-2.49e8 + 9.09e7i)T + (5.82e17 - 4.88e17i)T^{2} \) |
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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
−14.08222096842030098228181470069, −12.95074358159069469452867838780, −11.50632325793597611869874498876, −10.40793366495384690773490304747, −9.544452315483640844199131774727, −7.46119537283094618619566833532, −5.65367966423298509582175252651, −4.26324942961302149334121778761, −3.36636316978475115739236389368, −0.61288304638150014528694819854,
1.66098370497479825042401442933, 3.27521413454542345325791626278, 5.46714614551798411532784142134, 6.66798022460679492423711487970, 7.63877265483117436829652027285, 9.385601103219871664050645188302, 11.51000147451693712371118698197, 12.33039249864582465709914541114, 13.08872755289311922189000673095, 14.44927787468311292159248018239
