| L(s) = 1 | + (−0.284 + 1.97i)2-s + (0.934 + 2.04i)3-s + (−3.83 − 1.12i)4-s + (−3.27 − 3.77i)5-s + (−4.31 + 1.26i)6-s + (13.5 + 8.70i)7-s + (3.32 − 7.27i)8-s + (14.3 − 16.5i)9-s + (8.41 − 5.40i)10-s + (−6.55 − 45.5i)11-s + (−1.28 − 8.90i)12-s + (0.196 − 0.126i)13-s + (−21.0 + 24.3i)14-s + (4.67 − 10.2i)15-s + (13.4 + 8.65i)16-s + (41.4 − 12.1i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 + 0.699i)2-s + (0.179 + 0.393i)3-s + (−0.479 − 0.140i)4-s + (−0.292 − 0.337i)5-s + (−0.293 + 0.0862i)6-s + (0.731 + 0.469i)7-s + (0.146 − 0.321i)8-s + (0.532 − 0.614i)9-s + (0.266 − 0.170i)10-s + (−0.179 − 1.24i)11-s + (−0.0308 − 0.214i)12-s + (0.00419 − 0.00269i)13-s + (−0.402 + 0.464i)14-s + (0.0804 − 0.176i)15-s + (0.210 + 0.135i)16-s + (0.591 − 0.173i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.448i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.893 - 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.76304 + 0.417681i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.76304 + 0.417681i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.284 - 1.97i)T \) |
| 5 | \( 1 + (3.27 + 3.77i)T \) |
| 23 | \( 1 + (-109. - 15.6i)T \) |
| good | 3 | \( 1 + (-0.934 - 2.04i)T + (-17.6 + 20.4i)T^{2} \) |
| 7 | \( 1 + (-13.5 - 8.70i)T + (142. + 312. i)T^{2} \) |
| 11 | \( 1 + (6.55 + 45.5i)T + (-1.27e3 + 374. i)T^{2} \) |
| 13 | \( 1 + (-0.196 + 0.126i)T + (912. - 1.99e3i)T^{2} \) |
| 17 | \( 1 + (-41.4 + 12.1i)T + (4.13e3 - 2.65e3i)T^{2} \) |
| 19 | \( 1 + (7.67 + 2.25i)T + (5.77e3 + 3.70e3i)T^{2} \) |
| 29 | \( 1 + (-200. + 58.7i)T + (2.05e4 - 1.31e4i)T^{2} \) |
| 31 | \( 1 + (24.7 - 54.1i)T + (-1.95e4 - 2.25e4i)T^{2} \) |
| 37 | \( 1 + (57.5 - 66.4i)T + (-7.20e3 - 5.01e4i)T^{2} \) |
| 41 | \( 1 + (127. + 146. i)T + (-9.80e3 + 6.82e4i)T^{2} \) |
| 43 | \( 1 + (-195. - 427. i)T + (-5.20e4 + 6.00e4i)T^{2} \) |
| 47 | \( 1 - 57.3T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-223. - 143. i)T + (6.18e4 + 1.35e5i)T^{2} \) |
| 59 | \( 1 + (-169. + 108. i)T + (8.53e4 - 1.86e5i)T^{2} \) |
| 61 | \( 1 + (-301. + 660. i)T + (-1.48e5 - 1.71e5i)T^{2} \) |
| 67 | \( 1 + (-146. + 1.01e3i)T + (-2.88e5 - 8.47e4i)T^{2} \) |
| 71 | \( 1 + (129. - 902. i)T + (-3.43e5 - 1.00e5i)T^{2} \) |
| 73 | \( 1 + (471. + 138. i)T + (3.27e5 + 2.10e5i)T^{2} \) |
| 79 | \( 1 + (376. - 241. i)T + (2.04e5 - 4.48e5i)T^{2} \) |
| 83 | \( 1 + (-163. + 188. i)T + (-8.13e4 - 5.65e5i)T^{2} \) |
| 89 | \( 1 + (-347. - 760. i)T + (-4.61e5 + 5.32e5i)T^{2} \) |
| 97 | \( 1 + (855. + 987. i)T + (-1.29e5 + 9.03e5i)T^{2} \) |
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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
−11.84151940641320754828554729584, −10.81774788231777019890123456080, −9.640764984730560295637367128053, −8.696658344669468649265506743195, −8.073702279985026444628540808874, −6.78343589858028160831973778019, −5.56431729383457054905692148330, −4.59104440488775926063364082220, −3.25903719578294802390441022922, −0.947737824544055894266029810034,
1.27922024144792226726971273248, 2.51091411038701101621697343586, 4.13500106010023315408611458425, 5.07033290511374063776058307743, 7.02462796385732206457465506467, 7.65127382800183064810321398359, 8.701263807901973737007338035055, 10.14047785249184865216341300957, 10.58161974143278610349864098248, 11.72743821810312945755577180109
