| L(s) = 1 | + (−2.04 − 1.95i)2-s + (−2.83 + 4.35i)3-s + (0.328 + 7.99i)4-s + (−6.07 + 16.6i)5-s + (14.3 − 3.33i)6-s + (−22.8 − 4.02i)7-s + (14.9 − 16.9i)8-s + (−10.9 − 24.6i)9-s + (45.0 − 22.1i)10-s + (26.9 − 9.81i)11-s + (−35.7 − 21.2i)12-s + (34.3 − 28.8i)13-s + (38.6 + 52.8i)14-s + (−55.4 − 73.7i)15-s + (−63.7 + 5.25i)16-s + (19.0 − 10.9i)17-s + ⋯ |
| L(s) = 1 | + (−0.721 − 0.692i)2-s + (−0.545 + 0.838i)3-s + (0.0410 + 0.999i)4-s + (−0.543 + 1.49i)5-s + (0.973 − 0.226i)6-s + (−1.23 − 0.217i)7-s + (0.662 − 0.749i)8-s + (−0.404 − 0.914i)9-s + (1.42 − 0.700i)10-s + (0.739 − 0.269i)11-s + (−0.859 − 0.510i)12-s + (0.732 − 0.614i)13-s + (0.738 + 1.00i)14-s + (−0.954 − 1.26i)15-s + (−0.996 + 0.0821i)16-s + (0.271 − 0.156i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.644 + 0.764i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.644 + 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0310760 - 0.0668866i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0310760 - 0.0668866i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.04 + 1.95i)T \) |
| 3 | \( 1 + (2.83 - 4.35i)T \) |
| good | 5 | \( 1 + (6.07 - 16.6i)T + (-95.7 - 80.3i)T^{2} \) |
| 7 | \( 1 + (22.8 + 4.02i)T + (322. + 117. i)T^{2} \) |
| 11 | \( 1 + (-26.9 + 9.81i)T + (1.01e3 - 855. i)T^{2} \) |
| 13 | \( 1 + (-34.3 + 28.8i)T + (381. - 2.16e3i)T^{2} \) |
| 17 | \( 1 + (-19.0 + 10.9i)T + (2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (107. + 61.9i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-28.8 - 163. i)T + (-1.14e4 + 4.16e3i)T^{2} \) |
| 29 | \( 1 + (-19.0 + 22.6i)T + (-4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (29.1 - 5.14i)T + (2.79e4 - 1.01e4i)T^{2} \) |
| 37 | \( 1 + (99.4 + 172. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (305. + 364. i)T + (-1.19e4 + 6.78e4i)T^{2} \) |
| 43 | \( 1 + (-7.64 - 21.0i)T + (-6.09e4 + 5.11e4i)T^{2} \) |
| 47 | \( 1 + (0.135 - 0.765i)T + (-9.75e4 - 3.55e4i)T^{2} \) |
| 53 | \( 1 - 437. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (449. + 163. i)T + (1.57e5 + 1.32e5i)T^{2} \) |
| 61 | \( 1 + (-131. + 747. i)T + (-2.13e5 - 7.76e4i)T^{2} \) |
| 67 | \( 1 + (132. + 158. i)T + (-5.22e4 + 2.96e5i)T^{2} \) |
| 71 | \( 1 + (301. + 521. i)T + (-1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (372. - 644. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (345. - 411. i)T + (-8.56e4 - 4.85e5i)T^{2} \) |
| 83 | \( 1 + (515. + 432. i)T + (9.92e4 + 5.63e5i)T^{2} \) |
| 89 | \( 1 + (-609. - 351. i)T + (3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (1.08e3 - 393. i)T + (6.99e5 - 5.86e5i)T^{2} \) |
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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
−12.46522909373351359130324660532, −11.31085887562296777807848713703, −10.76823251146212729449512372804, −9.919922291103053940510584574036, −8.874023287448504182801131476077, −7.17631732078168152701481847061, −6.26592729942290666130963833558, −3.78051319291282342655311931433, −3.18811620662303073048905952985, −0.05853115622359570233494891463,
1.39841595733194261419916271446, 4.52971113745218420791791138094, 6.04646603025016693835443216576, 6.77522561300329081500192847534, 8.321388126731019596596320574402, 8.867787837176701917329331784638, 10.22393504114118067100029145654, 11.69769108634396469500336684260, 12.57886136443145252770784319829, 13.35353928139449321775000738007
