| L(s) = 1 | + (4.47 + 28.2i)2-s + (41.6 + 21.2i)3-s + (−535. + 173. i)4-s + (−597. + 181. i)5-s + (−413. + 1.27e3i)6-s + (1.19e3 + 1.19e3i)7-s + (−3.98e3 − 7.81e3i)8-s + (1.28e3 + 1.76e3i)9-s + (−7.81e3 − 1.60e4i)10-s + (−1.81e4 − 1.31e4i)11-s + (−2.59e4 − 4.11e3i)12-s + (84.9 − 536. i)13-s + (−2.83e4 + 3.90e4i)14-s + (−2.87e4 − 5.12e3i)15-s + (8.65e4 − 6.29e4i)16-s + (3.51e4 − 1.79e4i)17-s + ⋯ |
| L(s) = 1 | + (0.279 + 1.76i)2-s + (0.514 + 0.262i)3-s + (−2.09 + 0.679i)4-s + (−0.956 + 0.290i)5-s + (−0.319 + 0.981i)6-s + (0.496 + 0.496i)7-s + (−0.972 − 1.90i)8-s + (0.195 + 0.269i)9-s + (−0.781 − 1.60i)10-s + (−1.24 − 0.901i)11-s + (−1.25 − 0.198i)12-s + (0.00297 − 0.0187i)13-s + (−0.738 + 1.01i)14-s + (−0.568 − 0.101i)15-s + (1.32 − 0.960i)16-s + (0.421 − 0.214i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.315i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.948 + 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.0327154 - 0.00529455i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0327154 - 0.00529455i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-41.6 - 21.2i)T \) |
| 5 | \( 1 + (597. - 181. i)T \) |
| good | 2 | \( 1 + (-4.47 - 28.2i)T + (-243. + 79.1i)T^{2} \) |
| 7 | \( 1 + (-1.19e3 - 1.19e3i)T + 5.76e6iT^{2} \) |
| 11 | \( 1 + (1.81e4 + 1.31e4i)T + (6.62e7 + 2.03e8i)T^{2} \) |
| 13 | \( 1 + (-84.9 + 536. i)T + (-7.75e8 - 2.52e8i)T^{2} \) |
| 17 | \( 1 + (-3.51e4 + 1.79e4i)T + (4.10e9 - 5.64e9i)T^{2} \) |
| 19 | \( 1 + (1.65e4 + 5.37e3i)T + (1.37e10 + 9.98e9i)T^{2} \) |
| 23 | \( 1 + (2.12e5 - 3.36e4i)T + (7.44e10 - 2.41e10i)T^{2} \) |
| 29 | \( 1 + (-1.12e6 + 3.66e5i)T + (4.04e11 - 2.94e11i)T^{2} \) |
| 31 | \( 1 + (3.22e5 - 9.94e5i)T + (-6.90e11 - 5.01e11i)T^{2} \) |
| 37 | \( 1 + (1.74e6 + 2.76e5i)T + (3.34e12 + 1.08e12i)T^{2} \) |
| 41 | \( 1 + (-9.45e5 + 6.87e5i)T + (2.46e12 - 7.59e12i)T^{2} \) |
| 43 | \( 1 + (4.26e6 - 4.26e6i)T - 1.16e13iT^{2} \) |
| 47 | \( 1 + (-3.11e6 + 6.11e6i)T + (-1.39e13 - 1.92e13i)T^{2} \) |
| 53 | \( 1 + (1.24e6 + 6.32e5i)T + (3.65e13 + 5.03e13i)T^{2} \) |
| 59 | \( 1 + (-5.90e6 - 8.13e6i)T + (-4.53e13 + 1.39e14i)T^{2} \) |
| 61 | \( 1 + (-2.23e5 - 1.62e5i)T + (5.92e13 + 1.82e14i)T^{2} \) |
| 67 | \( 1 + (3.29e7 - 1.68e7i)T + (2.38e14 - 3.28e14i)T^{2} \) |
| 71 | \( 1 + (1.04e7 + 3.20e7i)T + (-5.22e14 + 3.79e14i)T^{2} \) |
| 73 | \( 1 + (1.38e7 - 2.18e6i)T + (7.66e14 - 2.49e14i)T^{2} \) |
| 79 | \( 1 + (4.11e7 - 1.33e7i)T + (1.22e15 - 8.91e14i)T^{2} \) |
| 83 | \( 1 + (2.64e7 + 5.19e7i)T + (-1.32e15 + 1.82e15i)T^{2} \) |
| 89 | \( 1 + (7.22e6 - 9.94e6i)T + (-1.21e15 - 3.74e15i)T^{2} \) |
| 97 | \( 1 + (-4.74e7 + 9.31e7i)T + (-4.60e15 - 6.34e15i)T^{2} \) |
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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
−14.26906454315880445167713217558, −13.37103085339015095031628611810, −11.96371610383688531813561722064, −10.38728802785847898456303841504, −8.563152010399401141758455687041, −8.194399365065994083332188377390, −7.12417539418903799792799441403, −5.64133773187369475998986679205, −4.57052238104313422441636685002, −3.13355692168743408415082886698,
0.009231185616261192816265239717, 1.37569036980567597774571330917, 2.68627784842103895996905109339, 3.97968913208591374838500640246, 4.90999257699543136799975345086, 7.54544385348395586175098275089, 8.535103213209189307410936946101, 9.986829030709691824795329011108, 10.79438793727908976502118827403, 12.00783913738626669355003611998
