| L(s) = 1 | + (3.80 + 1.51i)2-s + (26.7 + 2.91i)3-s + (−34.2 − 32.4i)4-s + (−125. + 148. i)5-s + (97.5 + 51.7i)6-s + (−29.7 − 17.9i)7-s + (−191. − 413. i)8-s + (−2.40 − 0.529i)9-s + (−704. + 373. i)10-s + (−1.82e3 − 99.2i)11-s + (−823. − 970. i)12-s + (−176. − 804. i)13-s + (−86.1 − 113. i)14-s + (−3.80e3 + 3.60e3i)15-s + (62.5 + 1.15e3i)16-s + (2.32e3 − 1.40e3i)17-s + ⋯ |
| L(s) = 1 | + (0.475 + 0.189i)2-s + (0.992 + 0.107i)3-s + (−0.535 − 0.507i)4-s + (−1.00 + 1.18i)5-s + (0.451 + 0.239i)6-s + (−0.0868 − 0.0522i)7-s + (−0.373 − 0.807i)8-s + (−0.00329 − 0.000726i)9-s + (−0.704 + 0.373i)10-s + (−1.37 − 0.0745i)11-s + (−0.476 − 0.561i)12-s + (−0.0805 − 0.365i)13-s + (−0.0313 − 0.0412i)14-s + (−1.12 + 1.06i)15-s + (0.0152 + 0.281i)16-s + (0.473 − 0.284i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0841i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.996 + 0.0841i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.0120787 - 0.286600i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0120787 - 0.286600i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 59 | \( 1 + (-7.82e4 - 1.89e5i)T \) |
| good | 2 | \( 1 + (-3.80 - 1.51i)T + (46.4 + 44.0i)T^{2} \) |
| 3 | \( 1 + (-26.7 - 2.91i)T + (711. + 156. i)T^{2} \) |
| 5 | \( 1 + (125. - 148. i)T + (-2.52e3 - 1.54e4i)T^{2} \) |
| 7 | \( 1 + (29.7 + 17.9i)T + (5.51e4 + 1.03e5i)T^{2} \) |
| 11 | \( 1 + (1.82e3 + 99.2i)T + (1.76e6 + 1.91e5i)T^{2} \) |
| 13 | \( 1 + (176. + 804. i)T + (-4.38e6 + 2.02e6i)T^{2} \) |
| 17 | \( 1 + (-2.32e3 + 1.40e3i)T + (1.13e7 - 2.13e7i)T^{2} \) |
| 19 | \( 1 + (1.77e3 - 6.37e3i)T + (-4.03e7 - 2.42e7i)T^{2} \) |
| 23 | \( 1 + (5.26e3 - 3.56e3i)T + (5.47e7 - 1.37e8i)T^{2} \) |
| 29 | \( 1 + (-6.05e3 - 1.51e4i)T + (-4.31e8 + 4.09e8i)T^{2} \) |
| 31 | \( 1 + (1.02e4 - 2.85e3i)T + (7.60e8 - 4.57e8i)T^{2} \) |
| 37 | \( 1 + (-4.05e3 + 8.76e3i)T + (-1.66e9 - 1.95e9i)T^{2} \) |
| 41 | \( 1 + (-2.28e4 + 3.37e4i)T + (-1.75e9 - 4.41e9i)T^{2} \) |
| 43 | \( 1 + (-6.41e4 + 3.47e3i)T + (6.28e9 - 6.83e8i)T^{2} \) |
| 47 | \( 1 + (-1.70e4 + 1.44e4i)T + (1.74e9 - 1.06e10i)T^{2} \) |
| 53 | \( 1 + (9.43e4 - 1.77e5i)T + (-1.24e10 - 1.83e10i)T^{2} \) |
| 61 | \( 1 + (-1.64e5 - 6.54e4i)T + (3.74e10 + 3.54e10i)T^{2} \) |
| 67 | \( 1 + (1.38e5 + 3.00e5i)T + (-5.85e10 + 6.89e10i)T^{2} \) |
| 71 | \( 1 + (-3.03e5 - 3.57e5i)T + (-2.07e10 + 1.26e11i)T^{2} \) |
| 73 | \( 1 + (-9.31e4 - 1.22e5i)T + (-4.04e10 + 1.45e11i)T^{2} \) |
| 79 | \( 1 + (1.66e5 - 1.81e4i)T + (2.37e11 - 5.22e10i)T^{2} \) |
| 83 | \( 1 + (2.77e5 + 8.24e5i)T + (-2.60e11 + 1.97e11i)T^{2} \) |
| 89 | \( 1 + (9.39e5 - 3.74e5i)T + (3.60e11 - 3.41e11i)T^{2} \) |
| 97 | \( 1 + (6.93e5 - 9.12e5i)T + (-2.22e11 - 8.02e11i)T^{2} \) |
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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
−14.51412158167275405989269350421, −13.74831348616399756010787941318, −12.46562599330831346547119562555, −10.85444433128089811330641563611, −9.900101422500633841378818028640, −8.350812032342961203767352745648, −7.37218448128994847123508485946, −5.65937799827426863587510132785, −3.87244146543973356310654583697, −2.86890699525019101656253224030,
0.087060008746063258468802699876, 2.66224526129260916175381458801, 4.03830390245798747729318852242, 5.15148339324423241759456046642, 7.85003655827736469972696227418, 8.302097337345944054118396143889, 9.363857628783182190394654186203, 11.37770643007265864696560078248, 12.58639570806949117343393349073, 13.13342440787641160204612906555
