| L(s) = 1 | + (9.74 + 16.8i)5-s + (18.4 − 1.67i)7-s + (−15.1 − 8.72i)11-s − 16.5i·13-s + (−68.6 + 118. i)17-s + (20.6 − 11.9i)19-s + (−108. + 62.8i)23-s + (−127. + 220. i)25-s − 105. i·29-s + (95.0 + 54.8i)31-s + (207. + 294. i)35-s + (58.5 + 101. i)37-s − 348.·41-s + 141.·43-s + (172. + 299. i)47-s + ⋯ |
| L(s) = 1 | + (0.871 + 1.50i)5-s + (0.995 − 0.0902i)7-s + (−0.414 − 0.239i)11-s − 0.352i·13-s + (−0.979 + 1.69i)17-s + (0.249 − 0.144i)19-s + (−0.987 + 0.570i)23-s + (−1.01 + 1.76i)25-s − 0.673i·29-s + (0.550 + 0.318i)31-s + (1.00 + 1.42i)35-s + (0.260 + 0.450i)37-s − 1.32·41-s + 0.503·43-s + (0.536 + 0.929i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.739 - 0.672i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.739 - 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.958342124\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.958342124\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-18.4 + 1.67i)T \) |
| good | 5 | \( 1 + (-9.74 - 16.8i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (15.1 + 8.72i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 16.5iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (68.6 - 118. i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-20.6 + 11.9i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (108. - 62.8i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 105. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (-95.0 - 54.8i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-58.5 - 101. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + 348.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 141.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-172. - 299. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (149. + 86.1i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (297. - 515. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (561. - 324. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-64.9 + 112. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 908. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (44.8 + 25.9i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (474. + 822. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 - 1.30e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-210. - 364. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 553. iT - 9.12e5T^{2} \) |
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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
−10.30456759290785859766800224139, −9.106706114154660930533335340029, −8.083434851010899250964378297966, −7.46110446140765796990139209724, −6.28713402670631028579910098280, −5.92959321345719697516971231880, −4.67491366961401238716899294690, −3.51125252238908143911184112334, −2.42831072193903535170310762593, −1.61489085792435193867550384943,
0.44620134882070912286328851996, 1.63378588418594680220184290243, 2.44614095772142453507590581017, 4.31101124649296237494015651975, 4.93487428205915463548323165992, 5.51736599933311097552092503431, 6.68023014617763036939764165553, 7.78992890543460188697111979689, 8.560850270837822767660771652808, 9.209506707465412097257970818135
