| L(s) = 1 | + (4.02 + 0.351i)2-s + (4.36 + 2.81i)3-s + (8.15 + 1.43i)4-s + (0.330 − 11.1i)5-s + (16.5 + 12.8i)6-s + (17.6 + 12.3i)7-s + (1.10 + 0.297i)8-s + (11.1 + 24.5i)9-s + (5.25 − 44.8i)10-s + (−10.8 + 29.7i)11-s + (31.5 + 29.2i)12-s + (−5.72 − 65.4i)13-s + (66.5 + 55.8i)14-s + (32.9 − 47.8i)15-s + (−57.9 − 21.0i)16-s + (−2.49 + 0.668i)17-s + ⋯ |
| L(s) = 1 | + (1.42 + 0.124i)2-s + (0.840 + 0.541i)3-s + (1.01 + 0.179i)4-s + (0.0295 − 0.999i)5-s + (1.12 + 0.874i)6-s + (0.952 + 0.667i)7-s + (0.0490 + 0.0131i)8-s + (0.412 + 0.910i)9-s + (0.166 − 1.41i)10-s + (−0.296 + 0.815i)11-s + (0.759 + 0.703i)12-s + (−0.122 − 1.39i)13-s + (1.27 + 1.06i)14-s + (0.566 − 0.824i)15-s + (−0.905 − 0.329i)16-s + (−0.0356 + 0.00954i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 - 0.366i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.930 - 0.366i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.28212 + 0.812516i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.28212 + 0.812516i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-4.36 - 2.81i)T \) |
| 5 | \( 1 + (-0.330 + 11.1i)T \) |
| good | 2 | \( 1 + (-4.02 - 0.351i)T + (7.87 + 1.38i)T^{2} \) |
| 7 | \( 1 + (-17.6 - 12.3i)T + (117. + 322. i)T^{2} \) |
| 11 | \( 1 + (10.8 - 29.7i)T + (-1.01e3 - 855. i)T^{2} \) |
| 13 | \( 1 + (5.72 + 65.4i)T + (-2.16e3 + 381. i)T^{2} \) |
| 17 | \( 1 + (2.49 - 0.668i)T + (4.25e3 - 2.45e3i)T^{2} \) |
| 19 | \( 1 + (-31.4 + 18.1i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (2.91 + 4.15i)T + (-4.16e3 + 1.14e4i)T^{2} \) |
| 29 | \( 1 + (191. - 160. i)T + (4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (9.93 - 56.3i)T + (-2.79e4 - 1.01e4i)T^{2} \) |
| 37 | \( 1 + (24.2 + 90.6i)T + (-4.38e4 + 2.53e4i)T^{2} \) |
| 41 | \( 1 + (-200. + 238. i)T + (-1.19e4 - 6.78e4i)T^{2} \) |
| 43 | \( 1 + (168. + 361. i)T + (-5.11e4 + 6.09e4i)T^{2} \) |
| 47 | \( 1 + (91.9 - 131. i)T + (-3.55e4 - 9.75e4i)T^{2} \) |
| 53 | \( 1 + (377. - 377. i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 + (-768. + 279. i)T + (1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (41.8 + 237. i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (-578. + 50.6i)T + (2.96e5 - 5.22e4i)T^{2} \) |
| 71 | \( 1 + (-24.2 - 13.9i)T + (1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (-104. + 391. i)T + (-3.36e5 - 1.94e5i)T^{2} \) |
| 79 | \( 1 + (-519. - 618. i)T + (-8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (111. - 1.27e3i)T + (-5.63e5 - 9.92e4i)T^{2} \) |
| 89 | \( 1 + (-322. - 557. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-1.26e3 + 588. i)T + (5.86e5 - 6.99e5i)T^{2} \) |
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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
−12.84093785741183157263324035298, −12.37067781610288899251939665782, −10.99525564919284198509637232825, −9.559976769090463589427798806604, −8.563637503072557765213832815423, −7.51639223594008389267859436512, −5.30869427095213105027543977854, −5.07251296277209685942767079829, −3.71625439426438497975074586686, −2.20618382152369892352255467164,
2.03369637666700630738722555374, 3.35651094458999574642958616059, 4.37496221808013028439291290786, 6.05926744194281454461897160956, 7.09001902988732548725163722446, 8.146277986244161032182983862754, 9.632665775389152033886302603600, 11.25816804969672592319726864747, 11.64899264121802670674639872892, 13.19371613682308643775279184209
