| L(s) = 1 | + (5.33 + 14.6i)3-s + (−7.24 + 41.0i)5-s + (−15.7 − 27.2i)7-s + (−124. + 104. i)9-s + (89.7 − 155. i)11-s + (34.6 − 95.2i)13-s + (−641. + 113. i)15-s + (−2.64 − 2.21i)17-s + (145. + 330. i)19-s + (315. − 375. i)21-s + (179. + 1.01e3i)23-s + (−1.04e3 − 380. i)25-s + (−1.10e3 − 637. i)27-s + (545. + 649. i)29-s + (−974. + 562. i)31-s + ⋯ |
| L(s) = 1 | + (0.593 + 1.62i)3-s + (−0.289 + 1.64i)5-s + (−0.320 − 0.555i)7-s + (−1.53 + 1.29i)9-s + (0.741 − 1.28i)11-s + (0.205 − 0.563i)13-s + (−2.84 + 0.502i)15-s + (−0.00914 − 0.00766i)17-s + (0.402 + 0.915i)19-s + (0.715 − 0.852i)21-s + (0.338 + 1.92i)23-s + (−1.67 − 0.609i)25-s + (−1.51 − 0.874i)27-s + (0.648 + 0.772i)29-s + (−1.01 + 0.585i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.897 - 0.441i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.897 - 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.379998 + 1.63243i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.379998 + 1.63243i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-145. - 330. i)T \) |
| good | 3 | \( 1 + (-5.33 - 14.6i)T + (-62.0 + 52.0i)T^{2} \) |
| 5 | \( 1 + (7.24 - 41.0i)T + (-587. - 213. i)T^{2} \) |
| 7 | \( 1 + (15.7 + 27.2i)T + (-1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-89.7 + 155. i)T + (-7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-34.6 + 95.2i)T + (-2.18e4 - 1.83e4i)T^{2} \) |
| 17 | \( 1 + (2.64 + 2.21i)T + (1.45e4 + 8.22e4i)T^{2} \) |
| 23 | \( 1 + (-179. - 1.01e3i)T + (-2.62e5 + 9.57e4i)T^{2} \) |
| 29 | \( 1 + (-545. - 649. i)T + (-1.22e5 + 6.96e5i)T^{2} \) |
| 31 | \( 1 + (974. - 562. i)T + (4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 - 89.0iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (344. + 946. i)T + (-2.16e6 + 1.81e6i)T^{2} \) |
| 43 | \( 1 + (83.5 - 473. i)T + (-3.21e6 - 1.16e6i)T^{2} \) |
| 47 | \( 1 + (-2.55e3 + 2.13e3i)T + (8.47e5 - 4.80e6i)T^{2} \) |
| 53 | \( 1 + (-819. + 144. i)T + (7.41e6 - 2.69e6i)T^{2} \) |
| 59 | \( 1 + (1.60e3 - 1.91e3i)T + (-2.10e6 - 1.19e7i)T^{2} \) |
| 61 | \( 1 + (369. + 2.09e3i)T + (-1.30e7 + 4.73e6i)T^{2} \) |
| 67 | \( 1 + (-1.20e3 - 1.44e3i)T + (-3.49e6 + 1.98e7i)T^{2} \) |
| 71 | \( 1 + (1.30e3 + 230. i)T + (2.38e7 + 8.69e6i)T^{2} \) |
| 73 | \( 1 + (-278. + 101. i)T + (2.17e7 - 1.82e7i)T^{2} \) |
| 79 | \( 1 + (-3.93e3 - 1.08e4i)T + (-2.98e7 + 2.50e7i)T^{2} \) |
| 83 | \( 1 + (4.76e3 + 8.24e3i)T + (-2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + (-3.53e3 + 9.70e3i)T + (-4.80e7 - 4.03e7i)T^{2} \) |
| 97 | \( 1 + (-2.26e3 + 2.70e3i)T + (-1.53e7 - 8.71e7i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.31486686499957640225069214561, −13.79259017283547868211060761847, −11.47920356095473944774991658143, −10.69611686464212773823633826304, −10.00369624451649168040599160761, −8.755675063002358505742619423246, −7.30894853288925569351109606378, −5.71151341300935396172011660421, −3.54087429791789022836086054676, −3.39106494310667823221598753978,
0.841932976454528754452423423687, 2.20242745446944029745259219122, 4.54269948300314218669945355197, 6.33800060782463127320484496461, 7.50382609010022953093667431420, 8.719803345411461538492290251728, 9.262132302866927949426432663638, 11.79135921990020957855416897905, 12.46499714712543242674805574780, 12.97311065568357048798508559915
