| L(s) = 1 | + (1.88 − 4.84i)3-s + (−3.20 − 5.55i)5-s + (−14.8 + 11.1i)7-s + (−19.8 − 18.2i)9-s + (16.2 + 9.39i)11-s − 3.82i·13-s + (−32.9 + 5.03i)15-s + (8.12 − 14.0i)17-s + (−129. + 74.9i)19-s + (25.8 + 92.6i)21-s + (−109. + 63.3i)23-s + (41.9 − 72.6i)25-s + (−126. + 61.6i)27-s + 168. i·29-s + (43.4 + 25.0i)31-s + ⋯ |
| L(s) = 1 | + (0.363 − 0.931i)3-s + (−0.286 − 0.496i)5-s + (−0.799 + 0.600i)7-s + (−0.735 − 0.676i)9-s + (0.446 + 0.257i)11-s − 0.0816i·13-s + (−0.567 + 0.0867i)15-s + (0.115 − 0.200i)17-s + (−1.56 + 0.904i)19-s + (0.268 + 0.963i)21-s + (−0.994 + 0.573i)23-s + (0.335 − 0.581i)25-s + (−0.898 + 0.439i)27-s + 1.07i·29-s + (0.251 + 0.145i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.388 - 0.921i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.388 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2388983951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2388983951\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.88 + 4.84i)T \) |
| 7 | \( 1 + (14.8 - 11.1i)T \) |
| good | 5 | \( 1 + (3.20 + 5.55i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (-16.2 - 9.39i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 3.82iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-8.12 + 14.0i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (129. - 74.9i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (109. - 63.3i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 168. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (-43.4 - 25.0i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-24.7 - 42.8i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 - 19.9T + 6.89e4T^{2} \) |
| 43 | \( 1 + 5.14T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-308. - 534. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (242. + 140. i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (274. - 475. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (507. - 293. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-378. + 655. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 351. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (530. + 306. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (505. + 876. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 1.01e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (178. + 309. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 - 228. iT - 9.12e5T^{2} \) |
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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
−11.81668708824047869719167282488, −10.44528378374672615126211983163, −9.281550040724017761890735549920, −8.596084778896506144198972930299, −7.69969696699191949517726218473, −6.54065683536998717286528191688, −5.84086651002982286430939832830, −4.21214816277492874832209066235, −2.91473633007424643269969868993, −1.58572890752706114510013874404,
0.07786349477621435097713566220, 2.51854263445196633433353309479, 3.70831644502641829727195589206, 4.41610457171059102633407905149, 5.99876333894754682602668064701, 6.92128657113757504262327636109, 8.150787009289183903727900029400, 9.065343583683518948203060664912, 10.00173418719474756286122877592, 10.69800877010746628852667819398
