| L(s) = 1 | + (0.326 − 1.00i)2-s + (−7.10 − 1.50i)3-s + (24.9 + 18.1i)4-s + (−45.3 − 78.6i)5-s + (−3.83 + 6.63i)6-s + (143. − 63.8i)7-s + (53.6 − 39.0i)8-s + (−173. − 77.3i)9-s + (−93.7 + 19.9i)10-s + (68.8 − 654. i)11-s + (−150. − 166. i)12-s + (−39.0 + 43.3i)13-s + (−17.3 − 164. i)14-s + (203. + 626. i)15-s + (283. + 873. i)16-s + (27.7 + 264. i)17-s + ⋯ |
| L(s) = 1 | + (0.0576 − 0.177i)2-s + (−0.455 − 0.0968i)3-s + (0.780 + 0.567i)4-s + (−0.811 − 1.40i)5-s + (−0.0434 + 0.0752i)6-s + (1.10 − 0.492i)7-s + (0.296 − 0.215i)8-s + (−0.715 − 0.318i)9-s + (−0.296 + 0.0629i)10-s + (0.171 − 1.63i)11-s + (−0.300 − 0.334i)12-s + (−0.0641 + 0.0712i)13-s + (−0.0235 − 0.224i)14-s + (0.233 + 0.719i)15-s + (0.277 + 0.852i)16-s + (0.0232 + 0.221i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0562 + 0.998i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0562 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.02135 - 0.965428i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.02135 - 0.965428i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 + (312. + 5.34e3i)T \) |
| good | 2 | \( 1 + (-0.326 + 1.00i)T + (-25.8 - 18.8i)T^{2} \) |
| 3 | \( 1 + (7.10 + 1.50i)T + (221. + 98.8i)T^{2} \) |
| 5 | \( 1 + (45.3 + 78.6i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 7 | \( 1 + (-143. + 63.8i)T + (1.12e4 - 1.24e4i)T^{2} \) |
| 11 | \( 1 + (-68.8 + 654. i)T + (-1.57e5 - 3.34e4i)T^{2} \) |
| 13 | \( 1 + (39.0 - 43.3i)T + (-3.88e4 - 3.69e5i)T^{2} \) |
| 17 | \( 1 + (-27.7 - 264. i)T + (-1.38e6 + 2.95e5i)T^{2} \) |
| 19 | \( 1 + (-1.06e3 - 1.18e3i)T + (-2.58e5 + 2.46e6i)T^{2} \) |
| 23 | \( 1 + (750. - 545. i)T + (1.98e6 - 6.12e6i)T^{2} \) |
| 29 | \( 1 + (1.18e3 - 3.64e3i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 37 | \( 1 + (-4.59e3 + 7.96e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + (-8.14e3 + 1.73e3i)T + (1.05e8 - 4.71e7i)T^{2} \) |
| 43 | \( 1 + (6.71e3 + 7.45e3i)T + (-1.53e7 + 1.46e8i)T^{2} \) |
| 47 | \( 1 + (-7.14e3 - 2.19e4i)T + (-1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (1.99e4 + 8.89e3i)T + (2.79e8 + 3.10e8i)T^{2} \) |
| 59 | \( 1 + (-3.66e4 - 7.78e3i)T + (6.53e8 + 2.90e8i)T^{2} \) |
| 61 | \( 1 - 5.60e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (-2.25e4 - 3.90e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + (2.90e4 + 1.29e4i)T + (1.20e9 + 1.34e9i)T^{2} \) |
| 73 | \( 1 + (-706. + 6.72e3i)T + (-2.02e9 - 4.31e8i)T^{2} \) |
| 79 | \( 1 + (-1.20e3 - 1.15e4i)T + (-3.00e9 + 6.39e8i)T^{2} \) |
| 83 | \( 1 + (3.03e4 - 6.45e3i)T + (3.59e9 - 1.60e9i)T^{2} \) |
| 89 | \( 1 + (-5.90e4 - 4.28e4i)T + (1.72e9 + 5.31e9i)T^{2} \) |
| 97 | \( 1 + (8.14e4 + 5.91e4i)T + (2.65e9 + 8.16e9i)T^{2} \) |
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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
−16.08976758172267270299421207912, −14.32756448870238906156604743554, −12.76490334302293921561443047544, −11.60943925502474710604142364137, −11.19576495168411858645556861753, −8.618845938317898141236920150912, −7.77494698339337991850376307533, −5.64776172450641533249187794723, −3.83145617701341612649567384890, −0.963225656787542128510337929904,
2.41382313410670916486758077126, 5.02183973448806353392742134864, 6.71834366507205329648291708983, 7.79269093694490586332224732351, 10.18111003579255414070956946637, 11.36960777253032934388541146323, 11.76064645479817332528999225724, 14.37900317403688359989308050598, 14.93204660385109982081990728440, 15.81045194131690298855984394736
