| L(s) = 1 | + (−1.80 − 0.867i)2-s + (−6.26 + 7.85i)3-s + (2.49 + 3.12i)4-s + (−7.20 − 3.47i)5-s + (18.1 − 8.71i)6-s + (14.4 − 18.1i)7-s + (−1.78 − 7.79i)8-s + (−16.4 − 72.0i)9-s + (9.97 + 12.5i)10-s + (−8.88 + 38.9i)11-s − 40.1·12-s + (7.55 − 33.1i)13-s + (−41.8 + 20.1i)14-s + (72.4 − 34.8i)15-s + (−3.56 + 15.5i)16-s − 43.6·17-s + ⋯ |
| L(s) = 1 | + (−0.637 − 0.306i)2-s + (−1.20 + 1.51i)3-s + (0.311 + 0.390i)4-s + (−0.644 − 0.310i)5-s + (1.23 − 0.593i)6-s + (0.782 − 0.980i)7-s + (−0.0786 − 0.344i)8-s + (−0.609 − 2.66i)9-s + (0.315 + 0.395i)10-s + (−0.243 + 1.06i)11-s − 0.966·12-s + (0.161 − 0.706i)13-s + (−0.799 + 0.384i)14-s + (1.24 − 0.600i)15-s + (−0.0556 + 0.243i)16-s − 0.622·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.257 + 0.966i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.257 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.156052 - 0.203081i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.156052 - 0.203081i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.80 + 0.867i)T \) |
| 29 | \( 1 + (148. - 48.7i)T \) |
| good | 3 | \( 1 + (6.26 - 7.85i)T + (-6.00 - 26.3i)T^{2} \) |
| 5 | \( 1 + (7.20 + 3.47i)T + (77.9 + 97.7i)T^{2} \) |
| 7 | \( 1 + (-14.4 + 18.1i)T + (-76.3 - 334. i)T^{2} \) |
| 11 | \( 1 + (8.88 - 38.9i)T + (-1.19e3 - 577. i)T^{2} \) |
| 13 | \( 1 + (-7.55 + 33.1i)T + (-1.97e3 - 953. i)T^{2} \) |
| 17 | \( 1 + 43.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + (77.8 + 97.6i)T + (-1.52e3 + 6.68e3i)T^{2} \) |
| 23 | \( 1 + (-80.5 + 38.7i)T + (7.58e3 - 9.51e3i)T^{2} \) |
| 31 | \( 1 + (75.1 + 36.1i)T + (1.85e4 + 2.32e4i)T^{2} \) |
| 37 | \( 1 + (0.516 + 2.26i)T + (-4.56e4 + 2.19e4i)T^{2} \) |
| 41 | \( 1 + 351.T + 6.89e4T^{2} \) |
| 43 | \( 1 + (65.0 - 31.3i)T + (4.95e4 - 6.21e4i)T^{2} \) |
| 47 | \( 1 + (-91.4 + 400. i)T + (-9.35e4 - 4.50e4i)T^{2} \) |
| 53 | \( 1 + (72.2 + 34.7i)T + (9.28e4 + 1.16e5i)T^{2} \) |
| 59 | \( 1 - 79.3T + 2.05e5T^{2} \) |
| 61 | \( 1 + (9.87 - 12.3i)T + (-5.05e4 - 2.21e5i)T^{2} \) |
| 67 | \( 1 + (-99.3 - 435. i)T + (-2.70e5 + 1.30e5i)T^{2} \) |
| 71 | \( 1 + (149. - 654. i)T + (-3.22e5 - 1.55e5i)T^{2} \) |
| 73 | \( 1 + (-905. + 436. i)T + (2.42e5 - 3.04e5i)T^{2} \) |
| 79 | \( 1 + (-150. - 660. i)T + (-4.44e5 + 2.13e5i)T^{2} \) |
| 83 | \( 1 + (677. + 849. i)T + (-1.27e5 + 5.57e5i)T^{2} \) |
| 89 | \( 1 + (964. + 464. i)T + (4.39e5 + 5.51e5i)T^{2} \) |
| 97 | \( 1 + (542. + 680. i)T + (-2.03e5 + 8.89e5i)T^{2} \) |
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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
−15.02173196658743455241400429766, −12.78916413983587884452523755091, −11.49450644974257099476740810373, −10.84572458450192062087607298888, −10.02891796764434439667855696687, −8.674124762874791004669887718281, −7.00458181797286685338269415312, −4.95375467382088283982879172501, −4.04926000642082442764813951421, −0.25561087339310701701663994754,
1.80571330946331252171384324590, 5.44355282691547918112487305793, 6.45721323048009032396641329210, 7.70236380163822997738100854451, 8.580017480010433601033039913609, 11.03332960948611721080823561303, 11.38223410194649198947426491105, 12.44602195282813368867059863771, 13.77200121852043715902422647356, 15.16129339787657070556202619135
