| L(s) = 1 | + (5.47 − 3.97i)2-s + (20.3 + 2.13i)3-s + (−5.64 + 17.3i)4-s + (21.9 − 37.9i)5-s + (119. − 69.2i)6-s + (611. − 129. i)7-s + (171. + 529. i)8-s + (−303. − 64.4i)9-s + (−30.9 − 294. i)10-s + (385. − 347. i)11-s + (−152. + 341. i)12-s + (−426. − 957. i)13-s + (2.82e3 − 3.14e3i)14-s + (527. − 726. i)15-s + (2.09e3 + 1.52e3i)16-s + (3.57e3 + 3.22e3i)17-s + ⋯ |
| L(s) = 1 | + (0.683 − 0.496i)2-s + (0.753 + 0.0792i)3-s + (−0.0881 + 0.271i)4-s + (0.175 − 0.303i)5-s + (0.555 − 0.320i)6-s + (1.78 − 0.379i)7-s + (0.335 + 1.03i)8-s + (−0.415 − 0.0884i)9-s + (−0.0309 − 0.294i)10-s + (0.289 − 0.260i)11-s + (−0.0879 + 0.197i)12-s + (−0.193 − 0.435i)13-s + (1.03 − 1.14i)14-s + (0.156 − 0.215i)15-s + (0.512 + 0.372i)16-s + (0.728 + 0.655i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 + 0.352i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.935 + 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(3.02224 - 0.550732i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.02224 - 0.550732i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 + (7.45e3 - 2.88e4i)T \) |
| good | 2 | \( 1 + (-5.47 + 3.97i)T + (19.7 - 60.8i)T^{2} \) |
| 3 | \( 1 + (-20.3 - 2.13i)T + (713. + 151. i)T^{2} \) |
| 5 | \( 1 + (-21.9 + 37.9i)T + (-7.81e3 - 1.35e4i)T^{2} \) |
| 7 | \( 1 + (-611. + 129. i)T + (1.07e5 - 4.78e4i)T^{2} \) |
| 11 | \( 1 + (-385. + 347. i)T + (1.85e5 - 1.76e6i)T^{2} \) |
| 13 | \( 1 + (426. + 957. i)T + (-3.22e6 + 3.58e6i)T^{2} \) |
| 17 | \( 1 + (-3.57e3 - 3.22e3i)T + (2.52e6 + 2.40e7i)T^{2} \) |
| 19 | \( 1 + (1.02e4 + 4.55e3i)T + (3.14e7 + 3.49e7i)T^{2} \) |
| 23 | \( 1 + (1.07e4 - 3.50e3i)T + (1.19e8 - 8.70e7i)T^{2} \) |
| 29 | \( 1 + (1.34e4 + 1.85e4i)T + (-1.83e8 + 5.65e8i)T^{2} \) |
| 37 | \( 1 + (4.11e4 - 2.37e4i)T + (1.28e9 - 2.22e9i)T^{2} \) |
| 41 | \( 1 + (-2.64e3 - 2.52e4i)T + (-4.64e9 + 9.87e8i)T^{2} \) |
| 43 | \( 1 + (-2.55e4 + 5.74e4i)T + (-4.22e9 - 4.69e9i)T^{2} \) |
| 47 | \( 1 + (-2.63e4 - 1.91e4i)T + (3.33e9 + 1.02e10i)T^{2} \) |
| 53 | \( 1 + (-4.89e4 + 2.30e5i)T + (-2.02e10 - 9.01e9i)T^{2} \) |
| 59 | \( 1 + (-3.93e4 + 3.74e5i)T + (-4.12e10 - 8.76e9i)T^{2} \) |
| 61 | \( 1 - 7.10e4iT - 5.15e10T^{2} \) |
| 67 | \( 1 + (1.08e5 - 1.88e5i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 + (2.01e5 + 4.28e4i)T + (1.17e11 + 5.21e10i)T^{2} \) |
| 73 | \( 1 + (1.60e5 - 1.44e5i)T + (1.58e10 - 1.50e11i)T^{2} \) |
| 79 | \( 1 + (-3.91e5 - 3.52e5i)T + (2.54e10 + 2.41e11i)T^{2} \) |
| 83 | \( 1 + (-6.25e5 + 6.57e4i)T + (3.19e11 - 6.79e10i)T^{2} \) |
| 89 | \( 1 + (-5.62e5 - 1.82e5i)T + (4.02e11 + 2.92e11i)T^{2} \) |
| 97 | \( 1 + (1.56e5 - 4.82e5i)T + (-6.73e11 - 4.89e11i)T^{2} \) |
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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
−14.88381867926573431287941102463, −14.28826842315768095464590786834, −13.23701585396518569561091441620, −11.86833716438099621657511349948, −10.80533147661828218063625326351, −8.660246432914568590792032744952, −7.946500626372112440124790407716, −5.19752104028771075930306719330, −3.79387069054772141416882748671, −1.97368194477068699526073993122,
1.97775984312720947093521412674, 4.36902500962840766124388373669, 5.79466168384009823830073045286, 7.61349021257063081991833341233, 8.919172173967849349510173610322, 10.62564382089593143122501024839, 12.13302139494251346169761058834, 13.88366821815386209670476372203, 14.52918319183107380280818151147, 14.95590322153541652985186303124
