| L(s) = 1 | + (−3.75 − 0.662i)2-s + (−1.11 + 1.32i)3-s + (9.92 + 3.61i)4-s + (−0.758 + 0.276i)5-s + (5.06 − 4.24i)6-s + (−5.10 − 8.84i)7-s + (−21.6 − 12.5i)8-s + (−0.520 − 2.95i)9-s + (3.03 − 0.534i)10-s + (7.18 − 12.4i)11-s + (−15.8 + 9.14i)12-s + (−2.56 − 3.06i)13-s + (13.3 + 36.6i)14-s + (0.478 − 1.31i)15-s + (40.8 + 34.2i)16-s + (2.00 − 11.3i)17-s + ⋯ |
| L(s) = 1 | + (−1.87 − 0.331i)2-s + (−0.371 + 0.442i)3-s + (2.48 + 0.903i)4-s + (−0.151 + 0.0552i)5-s + (0.843 − 0.708i)6-s + (−0.729 − 1.26i)7-s + (−2.71 − 1.56i)8-s + (−0.0578 − 0.328i)9-s + (0.303 − 0.0534i)10-s + (0.653 − 1.13i)11-s + (−1.32 + 0.762i)12-s + (−0.197 − 0.235i)13-s + (0.951 + 2.61i)14-s + (0.0318 − 0.0875i)15-s + (2.55 + 2.14i)16-s + (0.118 − 0.669i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.309 + 0.950i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.309 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.183736 - 0.253102i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.183736 - 0.253102i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.11 - 1.32i)T \) |
| 19 | \( 1 + (13.0 + 13.8i)T \) |
| good | 2 | \( 1 + (3.75 + 0.662i)T + (3.75 + 1.36i)T^{2} \) |
| 5 | \( 1 + (0.758 - 0.276i)T + (19.1 - 16.0i)T^{2} \) |
| 7 | \( 1 + (5.10 + 8.84i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-7.18 + 12.4i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (2.56 + 3.06i)T + (-29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (-2.00 + 11.3i)T + (-271. - 98.8i)T^{2} \) |
| 23 | \( 1 + (-15.8 - 5.77i)T + (405. + 340. i)T^{2} \) |
| 29 | \( 1 + (34.6 - 6.10i)T + (790. - 287. i)T^{2} \) |
| 31 | \( 1 + (-32.6 + 18.8i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 11.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (24.1 - 28.8i)T + (-291. - 1.65e3i)T^{2} \) |
| 43 | \( 1 + (3.41 - 1.24i)T + (1.41e3 - 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-1.47 - 8.33i)T + (-2.07e3 + 755. i)T^{2} \) |
| 53 | \( 1 + (-13.6 + 37.6i)T + (-2.15e3 - 1.80e3i)T^{2} \) |
| 59 | \( 1 + (-28.3 - 4.99i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-7.10 - 2.58i)T + (2.85e3 + 2.39e3i)T^{2} \) |
| 67 | \( 1 + (63.7 - 11.2i)T + (4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (39.3 + 108. i)T + (-3.86e3 + 3.24e3i)T^{2} \) |
| 73 | \( 1 + (-80.3 - 67.4i)T + (925. + 5.24e3i)T^{2} \) |
| 79 | \( 1 + (-83.3 + 99.3i)T + (-1.08e3 - 6.14e3i)T^{2} \) |
| 83 | \( 1 + (11.0 + 19.1i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-46.0 - 54.8i)T + (-1.37e3 + 7.80e3i)T^{2} \) |
| 97 | \( 1 + (-77.6 - 13.6i)T + (8.84e3 + 3.21e3i)T^{2} \) |
| 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
−15.22447741426847896018557466606, −13.31725542436491428139505159689, −11.61539349442964083346566386204, −10.92467370513934809507109040653, −9.907569187242638615433136952206, −9.005856661002002899122105258339, −7.53401397195364340681952206612, −6.45416479065738115085512026870, −3.42254654488228486878120781027, −0.55189718791332664963820502579,
2.03807935022084940657218710572, 5.99310236036223009851454310243, 6.93702707255251924525942887598, 8.300232007390861610505057104548, 9.327840873520148605309821315055, 10.30936911030186567281913067434, 11.79244570191450452505625203264, 12.45686839761400014244659773090, 14.86996089814364610340899295340, 15.59407321616715088355595699065
