| L(s) = 1 | + (1.81 − 2.16i)2-s + (4.86 + 13.3i)3-s + (−1.38 − 7.87i)4-s + (1.39 − 7.93i)5-s + (37.8 + 13.7i)6-s + (43.6 + 75.5i)7-s + (−19.5 − 11.3i)8-s + (−92.9 + 78.0i)9-s + (−14.6 − 17.4i)10-s + (76.3 − 132. i)11-s + (98.5 − 56.9i)12-s + (8.37 − 23.0i)13-s + (242. + 42.8i)14-s + (112. − 19.8i)15-s + (−60.1 + 21.8i)16-s + (−315. − 264. i)17-s + ⋯ |
| L(s) = 1 | + (0.454 − 0.541i)2-s + (0.540 + 1.48i)3-s + (−0.0868 − 0.492i)4-s + (0.0559 − 0.317i)5-s + (1.05 + 0.382i)6-s + (0.890 + 1.54i)7-s + (−0.306 − 0.176i)8-s + (−1.14 + 0.963i)9-s + (−0.146 − 0.174i)10-s + (0.630 − 1.09i)11-s + (0.684 − 0.395i)12-s + (0.0495 − 0.136i)13-s + (1.23 + 0.218i)14-s + (0.501 − 0.0884i)15-s + (−0.234 + 0.0855i)16-s + (−1.09 − 0.915i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.854 - 0.519i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.854 - 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.08592 + 0.584774i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.08592 + 0.584774i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.81 + 2.16i)T \) |
| 19 | \( 1 + (187. + 308. i)T \) |
| good | 3 | \( 1 + (-4.86 - 13.3i)T + (-62.0 + 52.0i)T^{2} \) |
| 5 | \( 1 + (-1.39 + 7.93i)T + (-587. - 213. i)T^{2} \) |
| 7 | \( 1 + (-43.6 - 75.5i)T + (-1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-76.3 + 132. i)T + (-7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-8.37 + 23.0i)T + (-2.18e4 - 1.83e4i)T^{2} \) |
| 17 | \( 1 + (315. + 264. i)T + (1.45e4 + 8.22e4i)T^{2} \) |
| 23 | \( 1 + (-30.2 - 171. i)T + (-2.62e5 + 9.57e4i)T^{2} \) |
| 29 | \( 1 + (639. + 762. i)T + (-1.22e5 + 6.96e5i)T^{2} \) |
| 31 | \( 1 + (91.0 - 52.5i)T + (4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 + 1.35e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (-346. - 952. i)T + (-2.16e6 + 1.81e6i)T^{2} \) |
| 43 | \( 1 + (311. - 1.76e3i)T + (-3.21e6 - 1.16e6i)T^{2} \) |
| 47 | \( 1 + (-1.68e3 + 1.41e3i)T + (8.47e5 - 4.80e6i)T^{2} \) |
| 53 | \( 1 + (-4.17e3 + 736. i)T + (7.41e6 - 2.69e6i)T^{2} \) |
| 59 | \( 1 + (3.07e3 - 3.66e3i)T + (-2.10e6 - 1.19e7i)T^{2} \) |
| 61 | \( 1 + (-734. - 4.16e3i)T + (-1.30e7 + 4.73e6i)T^{2} \) |
| 67 | \( 1 + (1.98e3 + 2.36e3i)T + (-3.49e6 + 1.98e7i)T^{2} \) |
| 71 | \( 1 + (4.32e3 + 763. i)T + (2.38e7 + 8.69e6i)T^{2} \) |
| 73 | \( 1 + (3.59e3 - 1.30e3i)T + (2.17e7 - 1.82e7i)T^{2} \) |
| 79 | \( 1 + (2.58e3 + 7.10e3i)T + (-2.98e7 + 2.50e7i)T^{2} \) |
| 83 | \( 1 + (-2.65e3 - 4.59e3i)T + (-2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + (-979. + 2.69e3i)T + (-4.80e7 - 4.03e7i)T^{2} \) |
| 97 | \( 1 + (8.72e3 - 1.03e4i)T + (-1.53e7 - 8.71e7i)T^{2} \) |
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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
−15.32067756846802178242686314418, −14.73869636718056682316034218615, −13.44787954721986440105822554058, −11.69229909188270550390059573088, −10.94625600103027214947539967681, −9.141591545567624003553960167919, −8.807051497782025381124591818136, −5.58471100378881272057108392055, −4.43521621081734681839670054213, −2.68912282551312736298364544154,
1.70473936187571640598356049621, 4.19622748698141633134169533783, 6.66676280213125030685339246729, 7.32468919322797656674419921244, 8.488915584753071506196827707598, 10.71457585215506274373874040423, 12.30308500284418657558134801181, 13.27660436251136812276880317452, 14.24892275496420272316462010658, 14.84792089535154085311512738210
