| L(s) = 1 | + (−1.68 − 0.183i)2-s + (1.94 − 2.28i)3-s + (−4.99 − 1.10i)4-s + (3.49 + 2.65i)5-s + (−3.69 + 3.50i)6-s + (0.0781 − 0.196i)7-s + (21.1 + 7.11i)8-s + (−1.45 − 8.88i)9-s + (−5.41 − 5.12i)10-s + (21.6 − 10.0i)11-s + (−12.2 + 9.29i)12-s + (0.213 − 1.30i)13-s + (−0.167 + 0.316i)14-s + (12.8 − 2.83i)15-s + (2.83 + 1.31i)16-s + (−48.2 − 121. i)17-s + ⋯ |
| L(s) = 1 | + (−0.596 − 0.0649i)2-s + (0.373 − 0.440i)3-s + (−0.624 − 0.137i)4-s + (0.312 + 0.237i)5-s + (−0.251 + 0.238i)6-s + (0.00421 − 0.0105i)7-s + (0.932 + 0.314i)8-s + (−0.0539 − 0.328i)9-s + (−0.171 − 0.162i)10-s + (0.594 − 0.275i)11-s + (−0.293 + 0.223i)12-s + (0.00456 − 0.0278i)13-s + (−0.00320 + 0.00604i)14-s + (0.221 − 0.0487i)15-s + (0.0442 + 0.0204i)16-s + (−0.687 − 1.72i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.375 + 0.926i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.375 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.551871 - 0.819263i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.551871 - 0.819263i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.94 + 2.28i)T \) |
| 59 | \( 1 + (274. + 360. i)T \) |
| good | 2 | \( 1 + (1.68 + 0.183i)T + (7.81 + 1.71i)T^{2} \) |
| 5 | \( 1 + (-3.49 - 2.65i)T + (33.4 + 120. i)T^{2} \) |
| 7 | \( 1 + (-0.0781 + 0.196i)T + (-249. - 235. i)T^{2} \) |
| 11 | \( 1 + (-21.6 + 10.0i)T + (861. - 1.01e3i)T^{2} \) |
| 13 | \( 1 + (-0.213 + 1.30i)T + (-2.08e3 - 701. i)T^{2} \) |
| 17 | \( 1 + (48.2 + 121. i)T + (-3.56e3 + 3.37e3i)T^{2} \) |
| 19 | \( 1 + (10.2 + 15.0i)T + (-2.53e3 + 6.37e3i)T^{2} \) |
| 23 | \( 1 + (2.72 - 50.3i)T + (-1.20e4 - 1.31e3i)T^{2} \) |
| 29 | \( 1 + (294. - 32.0i)T + (2.38e4 - 5.24e3i)T^{2} \) |
| 31 | \( 1 + (-86.2 + 127. i)T + (-1.10e4 - 2.76e4i)T^{2} \) |
| 37 | \( 1 + (-358. + 120. i)T + (4.03e4 - 3.06e4i)T^{2} \) |
| 41 | \( 1 + (23.3 + 430. i)T + (-6.85e4 + 7.45e3i)T^{2} \) |
| 43 | \( 1 + (394. + 182. i)T + (5.14e4 + 6.05e4i)T^{2} \) |
| 47 | \( 1 + (-278. + 211. i)T + (2.77e4 - 1.00e5i)T^{2} \) |
| 53 | \( 1 + (298. - 283. i)T + (8.06e3 - 1.48e5i)T^{2} \) |
| 61 | \( 1 + (-404. - 44.0i)T + (2.21e5 + 4.87e4i)T^{2} \) |
| 67 | \( 1 + (-452. - 152. i)T + (2.39e5 + 1.82e5i)T^{2} \) |
| 71 | \( 1 + (-107. + 82.0i)T + (9.57e4 - 3.44e5i)T^{2} \) |
| 73 | \( 1 + (256. - 482. i)T + (-2.18e5 - 3.21e5i)T^{2} \) |
| 79 | \( 1 + (-582. - 685. i)T + (-7.97e4 + 4.86e5i)T^{2} \) |
| 83 | \( 1 + (343. - 206. i)T + (2.67e5 - 5.05e5i)T^{2} \) |
| 89 | \( 1 + (-941. + 102. i)T + (6.88e5 - 1.51e5i)T^{2} \) |
| 97 | \( 1 + (-523. - 986. i)T + (-5.12e5 + 7.55e5i)T^{2} \) |
| 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
−11.81454723667779183133710457526, −10.84096653380432183999027536119, −9.527218388113899053044933475992, −9.120804449192298022091258166234, −7.907565882075159891962370903354, −6.91434726102667728091400242431, −5.49547181930978992317520424232, −4.02605862418530139945729256004, −2.24236103695567807289433797099, −0.55363447543207044602777194403,
1.62438695135413193526562923185, 3.73296142713244152732204571629, 4.72455501901694136581660655411, 6.24652644472890420521050446114, 7.76931588404363337596384445214, 8.647609669191104594363096252227, 9.437818523443409316836241257775, 10.21730777528909171162516540040, 11.31964672224978492125808553185, 12.85836771579320541981272554926
