| L(s) = 1 | + (−2.77 − 15.7i)2-s + (−167. + 140. i)3-s + (−240. + 87.5i)4-s + (1.63e3 + 596. i)5-s + (2.68e3 + 2.25e3i)6-s + (1.26e3 − 2.18e3i)7-s + (2.04e3 + 3.54e3i)8-s + (4.90e3 − 2.78e4i)9-s + (4.84e3 − 2.74e4i)10-s + (−9.57e3 − 1.65e4i)11-s + (2.80e4 − 4.85e4i)12-s + (8.53e4 + 7.16e4i)13-s + (−3.80e4 − 1.38e4i)14-s + (−3.58e5 + 1.30e5i)15-s + (5.02e4 − 4.21e4i)16-s + (3.17e4 + 1.80e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 − 0.696i)2-s + (−1.19 + 1.00i)3-s + (−0.469 + 0.171i)4-s + (1.17 + 0.426i)5-s + (0.845 + 0.709i)6-s + (0.199 − 0.344i)7-s + (0.176 + 0.306i)8-s + (0.249 − 1.41i)9-s + (0.153 − 0.868i)10-s + (−0.197 − 0.341i)11-s + (0.390 − 0.675i)12-s + (0.829 + 0.695i)13-s + (−0.264 − 0.0962i)14-s + (−1.82 + 0.665i)15-s + (0.191 − 0.160i)16-s + (0.0922 + 0.523i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.405 - 0.914i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.405 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.483975 + 0.743798i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.483975 + 0.743798i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.77 + 15.7i)T \) |
| 19 | \( 1 + (3.41e5 - 4.53e5i)T \) |
| good | 3 | \( 1 + (167. - 140. i)T + (3.41e3 - 1.93e4i)T^{2} \) |
| 5 | \( 1 + (-1.63e3 - 596. i)T + (1.49e6 + 1.25e6i)T^{2} \) |
| 7 | \( 1 + (-1.26e3 + 2.18e3i)T + (-2.01e7 - 3.49e7i)T^{2} \) |
| 11 | \( 1 + (9.57e3 + 1.65e4i)T + (-1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 + (-8.53e4 - 7.16e4i)T + (1.84e9 + 1.04e10i)T^{2} \) |
| 17 | \( 1 + (-3.17e4 - 1.80e5i)T + (-1.11e11 + 4.05e10i)T^{2} \) |
| 23 | \( 1 + (-1.75e5 + 6.38e4i)T + (1.37e12 - 1.15e12i)T^{2} \) |
| 29 | \( 1 + (6.67e5 - 3.78e6i)T + (-1.36e13 - 4.96e12i)T^{2} \) |
| 31 | \( 1 + (1.82e6 - 3.15e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + 1.95e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + (5.59e6 - 4.69e6i)T + (5.68e13 - 3.22e14i)T^{2} \) |
| 43 | \( 1 + (3.37e7 + 1.22e7i)T + (3.85e14 + 3.23e14i)T^{2} \) |
| 47 | \( 1 + (4.34e6 - 2.46e7i)T + (-1.05e15 - 3.82e14i)T^{2} \) |
| 53 | \( 1 + (-6.02e7 + 2.19e7i)T + (2.52e15 - 2.12e15i)T^{2} \) |
| 59 | \( 1 + (-1.50e7 - 8.52e7i)T + (-8.14e15 + 2.96e15i)T^{2} \) |
| 61 | \( 1 + (-8.08e7 + 2.94e7i)T + (8.95e15 - 7.51e15i)T^{2} \) |
| 67 | \( 1 + (-2.22e7 + 1.26e8i)T + (-2.55e16 - 9.30e15i)T^{2} \) |
| 71 | \( 1 + (3.73e8 + 1.36e8i)T + (3.51e16 + 2.94e16i)T^{2} \) |
| 73 | \( 1 + (2.45e8 - 2.06e8i)T + (1.02e16 - 5.79e16i)T^{2} \) |
| 79 | \( 1 + (1.25e7 - 1.05e7i)T + (2.08e16 - 1.18e17i)T^{2} \) |
| 83 | \( 1 + (-2.48e8 + 4.30e8i)T + (-9.34e16 - 1.61e17i)T^{2} \) |
| 89 | \( 1 + (-7.81e8 - 6.55e8i)T + (6.08e16 + 3.45e17i)T^{2} \) |
| 97 | \( 1 + (-2.22e8 - 1.26e9i)T + (-7.14e17 + 2.60e17i)T^{2} \) |
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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
−14.55019856476241138419417431389, −13.39748438546783701977086403609, −11.89152749624831660418824772954, −10.59965907178860570468312903778, −10.36736872526290494636872430347, −8.909526116778871447013754176719, −6.38935931776304329912987562231, −5.24845061235935998894394426621, −3.73487170561961046444734377962, −1.57521323265927109532954428414,
0.40197764285285761909351054692, 1.80217591445932204507041078948, 5.17948665375523573389395305959, 5.92479936640821745541193236735, 7.06127550106767796931674392916, 8.644102304502102334165037435229, 10.18476562609136740256279325906, 11.64378386179772800910125154537, 13.01803226809751433656323589520, 13.51796527660137148411215499544
