| L(s) = 1 | + (−4.88 − 21.3i)2-s + (−49.8 − 46.2i)3-s + (27.4 − 13.2i)4-s + (614. − 92.6i)5-s + (−745. + 1.29e3i)6-s + (2.27e3 + 3.94e3i)7-s + (−7.42e3 − 9.30e3i)8-s + (−1.12e3 − 1.50e4i)9-s + (−4.98e3 − 1.26e4i)10-s + (7.74e4 + 3.72e4i)11-s + (−1.97e3 − 610. i)12-s + (5.79e4 − 1.47e5i)13-s + (7.33e4 − 6.80e4i)14-s + (−3.48e4 − 2.37e4i)15-s + (−1.53e5 + 1.92e5i)16-s + (1.05e5 + 1.58e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.215 − 0.945i)2-s + (−0.355 − 0.329i)3-s + (0.0535 − 0.0258i)4-s + (0.439 − 0.0662i)5-s + (−0.234 + 0.406i)6-s + (0.358 + 0.621i)7-s + (−0.640 − 0.803i)8-s + (−0.0571 − 0.763i)9-s + (−0.157 − 0.401i)10-s + (1.59 + 0.767i)11-s + (−0.0275 − 0.00849i)12-s + (0.562 − 1.43i)13-s + (0.510 − 0.473i)14-s + (−0.177 − 0.121i)15-s + (−0.584 + 0.732i)16-s + (0.306 + 0.0461i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.902 + 0.431i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.902 + 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.409084 - 1.80349i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.409084 - 1.80349i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-1.20e7 - 1.89e7i)T \) |
| good | 2 | \( 1 + (4.88 + 21.3i)T + (-461. + 222. i)T^{2} \) |
| 3 | \( 1 + (49.8 + 46.2i)T + (1.47e3 + 1.96e4i)T^{2} \) |
| 5 | \( 1 + (-614. + 92.6i)T + (1.86e6 - 5.75e5i)T^{2} \) |
| 7 | \( 1 + (-2.27e3 - 3.94e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (-7.74e4 - 3.72e4i)T + (1.47e9 + 1.84e9i)T^{2} \) |
| 13 | \( 1 + (-5.79e4 + 1.47e5i)T + (-7.77e9 - 7.21e9i)T^{2} \) |
| 17 | \( 1 + (-1.05e5 - 1.58e4i)T + (1.13e11 + 3.49e10i)T^{2} \) |
| 19 | \( 1 + (-4.26e4 + 5.68e5i)T + (-3.19e11 - 4.80e10i)T^{2} \) |
| 23 | \( 1 + (-7.08e4 + 4.82e4i)T + (6.58e11 - 1.67e12i)T^{2} \) |
| 29 | \( 1 + (4.14e5 - 3.84e5i)T + (1.08e12 - 1.44e13i)T^{2} \) |
| 31 | \( 1 + (2.79e6 + 8.63e5i)T + (2.18e13 + 1.48e13i)T^{2} \) |
| 37 | \( 1 + (4.79e6 - 8.29e6i)T + (-6.49e13 - 1.12e14i)T^{2} \) |
| 41 | \( 1 + (7.74e6 + 3.39e7i)T + (-2.94e14 + 1.42e14i)T^{2} \) |
| 47 | \( 1 + (1.05e7 - 5.09e6i)T + (6.97e14 - 8.74e14i)T^{2} \) |
| 53 | \( 1 + (2.84e7 + 7.24e7i)T + (-2.41e15 + 2.24e15i)T^{2} \) |
| 59 | \( 1 + (3.75e7 - 4.71e7i)T + (-1.92e15 - 8.44e15i)T^{2} \) |
| 61 | \( 1 + (-1.91e8 + 5.91e7i)T + (9.66e15 - 6.58e15i)T^{2} \) |
| 67 | \( 1 + (1.32e7 - 1.76e8i)T + (-2.69e16 - 4.05e15i)T^{2} \) |
| 71 | \( 1 + (-2.36e8 - 1.61e8i)T + (1.67e16 + 4.26e16i)T^{2} \) |
| 73 | \( 1 + (-9.43e7 + 2.40e8i)T + (-4.31e16 - 4.00e16i)T^{2} \) |
| 79 | \( 1 + (2.86e8 + 4.95e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + (2.09e8 + 1.94e8i)T + (1.39e16 + 1.86e17i)T^{2} \) |
| 89 | \( 1 + (-1.58e8 - 1.47e8i)T + (2.61e16 + 3.49e17i)T^{2} \) |
| 97 | \( 1 + (-6.70e8 - 3.22e8i)T + (4.73e17 + 5.94e17i)T^{2} \) |
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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
−12.97314564290802296021224352564, −12.06707165165185066730812921862, −11.30097428364011379474168001400, −9.884397678941905952786373241817, −8.925926981195471517240191562520, −6.84833438023614967540068305009, −5.70308895726133882591964196959, −3.51213229075412129376050720127, −1.85897096356414571161439941855, −0.75651132173108244646421863214,
1.65154869139060117780796045622, 4.00220871563950595360509831015, 5.74327070824930807825646191637, 6.73091901018086592684928035128, 8.110740970995083921576132217230, 9.380744170407284018437019112082, 11.00106955421469372194114859406, 11.79632517349502571463844845682, 13.96107655550605915795596732460, 14.31074385227395498199008278500
