| L(s) = 1 | + (1.17 − 1.61i)2-s + (−4.16 + 1.35i)3-s + (−1.23 − 3.80i)4-s + (−1.15 − 11.1i)5-s + (−2.70 + 8.33i)6-s − 31.2i·7-s + (−7.60 − 2.47i)8-s + (−6.32 + 4.59i)9-s + (−19.3 − 11.2i)10-s + (12.0 + 8.76i)11-s + (10.2 + 14.1i)12-s + (25.0 + 34.5i)13-s + (−50.4 − 36.6i)14-s + (19.8 + 44.7i)15-s + (−12.9 + 9.40i)16-s + (27.6 + 8.99i)17-s + ⋯ |
| L(s) = 1 | + (0.415 − 0.572i)2-s + (−0.801 + 0.260i)3-s + (−0.154 − 0.475i)4-s + (−0.103 − 0.994i)5-s + (−0.184 + 0.566i)6-s − 1.68i·7-s + (−0.336 − 0.109i)8-s + (−0.234 + 0.170i)9-s + (−0.611 − 0.354i)10-s + (0.330 + 0.240i)11-s + (0.247 + 0.340i)12-s + (0.535 + 0.736i)13-s + (−0.963 − 0.700i)14-s + (0.341 + 0.770i)15-s + (−0.202 + 0.146i)16-s + (0.394 + 0.128i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.519 + 0.854i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.519 + 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.556695 - 0.990617i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.556695 - 0.990617i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.17 + 1.61i)T \) |
| 5 | \( 1 + (1.15 + 11.1i)T \) |
| good | 3 | \( 1 + (4.16 - 1.35i)T + (21.8 - 15.8i)T^{2} \) |
| 7 | \( 1 + 31.2iT - 343T^{2} \) |
| 11 | \( 1 + (-12.0 - 8.76i)T + (411. + 1.26e3i)T^{2} \) |
| 13 | \( 1 + (-25.0 - 34.5i)T + (-678. + 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-27.6 - 8.99i)T + (3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-15.9 + 48.9i)T + (-5.54e3 - 4.03e3i)T^{2} \) |
| 23 | \( 1 + (-109. + 150. i)T + (-3.75e3 - 1.15e4i)T^{2} \) |
| 29 | \( 1 + (-23.7 - 73.1i)T + (-1.97e4 + 1.43e4i)T^{2} \) |
| 31 | \( 1 + (80.3 - 247. i)T + (-2.41e4 - 1.75e4i)T^{2} \) |
| 37 | \( 1 + (-70.7 - 97.3i)T + (-1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (-248. + 180. i)T + (2.12e4 - 6.55e4i)T^{2} \) |
| 43 | \( 1 + 504. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (186. - 60.7i)T + (8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-430. + 139. i)T + (1.20e5 - 8.75e4i)T^{2} \) |
| 59 | \( 1 + (-51.9 + 37.7i)T + (6.34e4 - 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-110. - 80.3i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 + (223. + 72.6i)T + (2.43e5 + 1.76e5i)T^{2} \) |
| 71 | \( 1 + (-166. - 512. i)T + (-2.89e5 + 2.10e5i)T^{2} \) |
| 73 | \( 1 + (614. - 846. i)T + (-1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (109. + 335. i)T + (-3.98e5 + 2.89e5i)T^{2} \) |
| 83 | \( 1 + (-928. - 301. i)T + (4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 + (-891. - 647. i)T + (2.17e5 + 6.70e5i)T^{2} \) |
| 97 | \( 1 + (1.37e3 - 448. i)T + (7.38e5 - 5.36e5i)T^{2} \) |
| 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
−14.33428857099000516310153572920, −13.42546604204681863407174562718, −12.29347847035359787837437714979, −11.14250132018808993337801337390, −10.33033760033676824295747095477, −8.785165692041225204562626614181, −6.86659767582721275261088564864, −5.09303715382088608351199770998, −4.07231650725225958200049384232, −0.896108432262587758517489864856,
3.11195252216297318117596230446, 5.64457931331132645947077438665, 6.16253298880257773947857267174, 7.80415226921888799297599988900, 9.327370089549621375523274561480, 11.24484384977717192742784777518, 11.87960707897581933739729068535, 13.10587701916879859469328590373, 14.67156291115286263171758103685, 15.23642701713143791676510569372
