| L(s) = 1 | + (1.87 + 0.684i)2-s + (−0.520 − 2.95i)3-s + (3.06 + 2.57i)4-s + (−11.3 + 9.51i)5-s + (1.04 − 5.90i)6-s + (−12.3 + 21.3i)7-s + (4.00 + 6.92i)8-s + (−8.45 + 3.07i)9-s + (−27.8 + 10.1i)10-s + (12.5 + 21.7i)11-s + (6.00 − 10.3i)12-s + (−0.865 + 4.91i)13-s + (−37.8 + 31.7i)14-s + (34.0 + 28.5i)15-s + (2.77 + 15.7i)16-s + (76.6 + 27.9i)17-s + ⋯ |
| L(s) = 1 | + (0.664 + 0.241i)2-s + (−0.100 − 0.568i)3-s + (0.383 + 0.321i)4-s + (−1.01 + 0.850i)5-s + (0.0708 − 0.402i)6-s + (−0.666 + 1.15i)7-s + (0.176 + 0.306i)8-s + (−0.313 + 0.114i)9-s + (−0.879 + 0.320i)10-s + (0.343 + 0.594i)11-s + (0.144 − 0.250i)12-s + (−0.0184 + 0.104i)13-s + (−0.721 + 0.605i)14-s + (0.585 + 0.491i)15-s + (0.0434 + 0.246i)16-s + (1.09 + 0.398i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.400 - 0.916i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.400 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.774067 + 1.18333i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.774067 + 1.18333i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.87 - 0.684i)T \) |
| 3 | \( 1 + (0.520 + 2.95i)T \) |
| 19 | \( 1 + (68.5 + 46.4i)T \) |
| good | 5 | \( 1 + (11.3 - 9.51i)T + (21.7 - 123. i)T^{2} \) |
| 7 | \( 1 + (12.3 - 21.3i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (-12.5 - 21.7i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (0.865 - 4.91i)T + (-2.06e3 - 751. i)T^{2} \) |
| 17 | \( 1 + (-76.6 - 27.9i)T + (3.76e3 + 3.15e3i)T^{2} \) |
| 23 | \( 1 + (5.93 + 4.97i)T + (2.11e3 + 1.19e4i)T^{2} \) |
| 29 | \( 1 + (136. - 49.6i)T + (1.86e4 - 1.56e4i)T^{2} \) |
| 31 | \( 1 + (-36.9 + 64.0i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + 171.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-29.7 - 168. i)T + (-6.47e4 + 2.35e4i)T^{2} \) |
| 43 | \( 1 + (-137. + 115. i)T + (1.38e4 - 7.82e4i)T^{2} \) |
| 47 | \( 1 + (-419. + 152. i)T + (7.95e4 - 6.67e4i)T^{2} \) |
| 53 | \( 1 + (-457. - 383. i)T + (2.58e4 + 1.46e5i)T^{2} \) |
| 59 | \( 1 + (-806. - 293. i)T + (1.57e5 + 1.32e5i)T^{2} \) |
| 61 | \( 1 + (-167. - 140. i)T + (3.94e4 + 2.23e5i)T^{2} \) |
| 67 | \( 1 + (667. - 242. i)T + (2.30e5 - 1.93e5i)T^{2} \) |
| 71 | \( 1 + (-108. + 90.6i)T + (6.21e4 - 3.52e5i)T^{2} \) |
| 73 | \( 1 + (207. + 1.17e3i)T + (-3.65e5 + 1.33e5i)T^{2} \) |
| 79 | \( 1 + (-179. - 1.02e3i)T + (-4.63e5 + 1.68e5i)T^{2} \) |
| 83 | \( 1 + (-405. + 702. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (50.8 - 288. i)T + (-6.62e5 - 2.41e5i)T^{2} \) |
| 97 | \( 1 + (463. + 168. i)T + (6.99e5 + 5.86e5i)T^{2} \) |
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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
−13.30724866802834646553539123832, −12.24617185317823270894838557182, −11.84599841005136134365547226960, −10.57391416807119809215296799477, −8.943326529450037045922703002908, −7.62312756530789716167012225349, −6.74432429723926861166189731568, −5.65434074411363247307785012268, −3.84533574883563298839408950246, −2.53349568046938197207957128358,
0.62507836551708120041968331241, 3.55433377487449203225976448644, 4.20427130589943451097811033967, 5.62207328617828800024708182735, 7.16555366539566326203503149395, 8.431524013463074095017442049195, 9.822366427590304220527281284171, 10.79784353145580057590330036426, 11.86334940994117870857089252169, 12.69367048646990312020125617510
