| L(s) = 1 | + (−4.06 − 0.355i)2-s + (5.18 − 0.359i)3-s + (8.51 + 1.50i)4-s + (10.6 + 3.28i)5-s + (−21.2 − 0.381i)6-s + (8.64 + 6.05i)7-s + (−2.56 − 0.687i)8-s + (26.7 − 3.72i)9-s + (−42.2 − 17.1i)10-s + (−16.2 + 44.6i)11-s + (44.7 + 4.72i)12-s + (−2.14 − 24.5i)13-s + (−32.9 − 27.6i)14-s + (56.5 + 13.2i)15-s + (−54.8 − 19.9i)16-s + (−103. + 27.7i)17-s + ⋯ |
| L(s) = 1 | + (−1.43 − 0.125i)2-s + (0.997 − 0.0692i)3-s + (1.06 + 0.187i)4-s + (0.955 + 0.294i)5-s + (−1.44 − 0.0259i)6-s + (0.466 + 0.326i)7-s + (−0.113 − 0.0304i)8-s + (0.990 − 0.138i)9-s + (−1.33 − 0.543i)10-s + (−0.445 + 1.22i)11-s + (1.07 + 0.113i)12-s + (−0.0458 − 0.523i)13-s + (−0.629 − 0.528i)14-s + (0.973 + 0.227i)15-s + (−0.856 − 0.311i)16-s + (−1.47 + 0.395i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.32296 + 0.310429i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.32296 + 0.310429i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-5.18 + 0.359i)T \) |
| 5 | \( 1 + (-10.6 - 3.28i)T \) |
| good | 2 | \( 1 + (4.06 + 0.355i)T + (7.87 + 1.38i)T^{2} \) |
| 7 | \( 1 + (-8.64 - 6.05i)T + (117. + 322. i)T^{2} \) |
| 11 | \( 1 + (16.2 - 44.6i)T + (-1.01e3 - 855. i)T^{2} \) |
| 13 | \( 1 + (2.14 + 24.5i)T + (-2.16e3 + 381. i)T^{2} \) |
| 17 | \( 1 + (103. - 27.7i)T + (4.25e3 - 2.45e3i)T^{2} \) |
| 19 | \( 1 + (-75.6 + 43.6i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-109. - 155. i)T + (-4.16e3 + 1.14e4i)T^{2} \) |
| 29 | \( 1 + (-23.0 + 19.3i)T + (4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (-32.5 + 184. i)T + (-2.79e4 - 1.01e4i)T^{2} \) |
| 37 | \( 1 + (-115. - 430. i)T + (-4.38e4 + 2.53e4i)T^{2} \) |
| 41 | \( 1 + (-103. + 123. i)T + (-1.19e4 - 6.78e4i)T^{2} \) |
| 43 | \( 1 + (118. + 255. i)T + (-5.11e4 + 6.09e4i)T^{2} \) |
| 47 | \( 1 + (-83.6 + 119. i)T + (-3.55e4 - 9.75e4i)T^{2} \) |
| 53 | \( 1 + (173. - 173. i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 + (-206. + 74.9i)T + (1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (66.4 + 376. i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (310. - 27.1i)T + (2.96e5 - 5.22e4i)T^{2} \) |
| 71 | \( 1 + (560. + 323. i)T + (1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (122. - 455. i)T + (-3.36e5 - 1.94e5i)T^{2} \) |
| 79 | \( 1 + (-513. - 612. i)T + (-8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (-43.5 + 497. i)T + (-5.63e5 - 9.92e4i)T^{2} \) |
| 89 | \( 1 + (159. + 277. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-713. + 332. i)T + (5.86e5 - 6.99e5i)T^{2} \) |
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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
−13.09898973175852156300934812241, −11.42886027129681408699027611393, −10.27214987888371617292553420434, −9.584617240158777278112190237140, −8.847597392156475250105231589796, −7.73909738302358303794641029352, −6.88683295356102126438889694612, −4.90315920137320379456482856897, −2.60035870521312496834185744973, −1.61674805378828717408796468211,
1.12217070607173217494065224572, 2.55270516800274526589344688274, 4.66468299145742565357244844211, 6.55788580857988741324922071769, 7.72202156840875283769778377320, 8.806824914338542393498362468370, 9.138026160112663288011596386304, 10.35427757572130601958961917935, 11.06862409075053690633042039711, 12.97198648344390444209573566334
