| L(s) = 1 | + (12.2 − 10.2i)2-s + (−169. − 61.7i)3-s + (44.4 − 252. i)4-s + (−248. − 1.40e3i)5-s + (−2.71e3 + 988. i)6-s + (5.31e3 − 9.19e3i)7-s + (−2.04e3 − 3.54e3i)8-s + (9.91e3 + 8.31e3i)9-s + (−1.75e4 − 1.46e4i)10-s + (−8.36e3 − 1.44e4i)11-s + (−2.31e4 + 4.00e4i)12-s + (8.37e3 − 3.04e3i)13-s + (−2.95e4 − 1.67e5i)14-s + (−4.48e4 + 2.54e5i)15-s + (−6.15e4 − 2.24e4i)16-s + (−1.13e5 + 9.49e4i)17-s + ⋯ |
| L(s) = 1 | + (0.541 − 0.454i)2-s + (−1.20 − 0.440i)3-s + (0.0868 − 0.492i)4-s + (−0.177 − 1.00i)5-s + (−0.855 + 0.311i)6-s + (0.836 − 1.44i)7-s + (−0.176 − 0.306i)8-s + (0.503 + 0.422i)9-s + (−0.553 − 0.464i)10-s + (−0.172 − 0.298i)11-s + (−0.321 + 0.557i)12-s + (0.0813 − 0.0295i)13-s + (−0.205 − 1.16i)14-s + (−0.228 + 1.29i)15-s + (−0.234 − 0.0855i)16-s + (−0.328 + 0.275i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.734 - 0.678i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.734 - 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.421368 + 1.07683i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.421368 + 1.07683i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-12.2 + 10.2i)T \) |
| 19 | \( 1 + (4.46e5 - 3.50e5i)T \) |
| good | 3 | \( 1 + (169. + 61.7i)T + (1.50e4 + 1.26e4i)T^{2} \) |
| 5 | \( 1 + (248. + 1.40e3i)T + (-1.83e6 + 6.68e5i)T^{2} \) |
| 7 | \( 1 + (-5.31e3 + 9.19e3i)T + (-2.01e7 - 3.49e7i)T^{2} \) |
| 11 | \( 1 + (8.36e3 + 1.44e4i)T + (-1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 + (-8.37e3 + 3.04e3i)T + (8.12e9 - 6.81e9i)T^{2} \) |
| 17 | \( 1 + (1.13e5 - 9.49e4i)T + (2.05e10 - 1.16e11i)T^{2} \) |
| 23 | \( 1 + (-6.02e4 + 3.41e5i)T + (-1.69e12 - 6.16e11i)T^{2} \) |
| 29 | \( 1 + (-3.37e6 - 2.83e6i)T + (2.51e12 + 1.42e13i)T^{2} \) |
| 31 | \( 1 + (-3.06e6 + 5.31e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + 2.06e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + (-1.75e7 - 6.38e6i)T + (2.50e14 + 2.10e14i)T^{2} \) |
| 43 | \( 1 + (-2.29e6 - 1.30e7i)T + (-4.72e14 + 1.71e14i)T^{2} \) |
| 47 | \( 1 + (5.43e6 + 4.55e6i)T + (1.94e14 + 1.10e15i)T^{2} \) |
| 53 | \( 1 + (-1.03e7 + 5.84e7i)T + (-3.10e15 - 1.12e15i)T^{2} \) |
| 59 | \( 1 + (-7.43e7 + 6.23e7i)T + (1.50e15 - 8.53e15i)T^{2} \) |
| 61 | \( 1 + (-4.04e6 + 2.29e7i)T + (-1.09e16 - 3.99e15i)T^{2} \) |
| 67 | \( 1 + (-1.89e8 - 1.59e8i)T + (4.72e15 + 2.67e16i)T^{2} \) |
| 71 | \( 1 + (2.58e7 + 1.46e8i)T + (-4.30e16 + 1.56e16i)T^{2} \) |
| 73 | \( 1 + (3.97e8 + 1.44e8i)T + (4.50e16 + 3.78e16i)T^{2} \) |
| 79 | \( 1 + (-3.43e8 - 1.25e8i)T + (9.18e16 + 7.70e16i)T^{2} \) |
| 83 | \( 1 + (8.19e7 - 1.41e8i)T + (-9.34e16 - 1.61e17i)T^{2} \) |
| 89 | \( 1 + (8.20e8 - 2.98e8i)T + (2.68e17 - 2.25e17i)T^{2} \) |
| 97 | \( 1 + (1.02e9 - 8.62e8i)T + (1.32e17 - 7.48e17i)T^{2} \) |
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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
−13.22962809729359891624269806248, −12.34231671203671417632937284880, −11.24369126480747158030375386502, −10.42281635078804451823847100872, −8.292424571939844230931977653879, −6.66825119052375299550823457754, −5.18707919313624575635807185207, −4.18139658664453919577527956123, −1.36724106255839862002922576013, −0.44040068631597593401652709700,
2.55792417237083747417748355344, 4.66454346365430107416825178333, 5.69074002573996207911435395041, 6.86262036523595888790820970149, 8.608570950051448888287713226313, 10.57337677032514683592962895588, 11.50337994919456026257506118096, 12.31308292914617006125886371288, 14.13714727961830584876698738367, 15.29629407182553299254715397488
