| L(s) = 1 | + (0.607 − 0.302i)2-s + (−0.927 + 1.22i)4-s + (0.905 + 0.825i)5-s + (−2.56 + 0.480i)7-s + (−0.441 + 2.36i)8-s + (0.800 + 0.227i)10-s + (−1.84 − 3.71i)11-s + (−1.91 + 0.358i)13-s + (−1.41 + 1.06i)14-s + (−0.395 − 1.38i)16-s + (−2.13 + 0.197i)17-s + (−2.31 + 1.43i)19-s + (−1.85 + 0.346i)20-s + (−2.24 − 1.69i)22-s + (1.96 + 0.975i)23-s + ⋯ |
| L(s) = 1 | + (0.429 − 0.214i)2-s + (−0.463 + 0.613i)4-s + (0.405 + 0.369i)5-s + (−0.971 + 0.181i)7-s + (−0.156 + 0.835i)8-s + (0.253 + 0.0720i)10-s + (−0.557 − 1.12i)11-s + (−0.531 + 0.0994i)13-s + (−0.378 + 0.285i)14-s + (−0.0988 − 0.347i)16-s + (−0.517 + 0.0479i)17-s + (−0.531 + 0.329i)19-s + (−0.414 + 0.0775i)20-s + (−0.479 − 0.362i)22-s + (0.408 + 0.203i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00876062 - 0.200697i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00876062 - 0.200697i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 103 | \( 1 + (4.41 + 9.13i)T \) |
| good | 2 | \( 1 + (-0.607 + 0.302i)T + (1.20 - 1.59i)T^{2} \) |
| 5 | \( 1 + (-0.905 - 0.825i)T + (0.461 + 4.97i)T^{2} \) |
| 7 | \( 1 + (2.56 - 0.480i)T + (6.52 - 2.52i)T^{2} \) |
| 11 | \( 1 + (1.84 + 3.71i)T + (-6.62 + 8.77i)T^{2} \) |
| 13 | \( 1 + (1.91 - 0.358i)T + (12.1 - 4.69i)T^{2} \) |
| 17 | \( 1 + (2.13 - 0.197i)T + (16.7 - 3.12i)T^{2} \) |
| 19 | \( 1 + (2.31 - 1.43i)T + (8.46 - 17.0i)T^{2} \) |
| 23 | \( 1 + (-1.96 - 0.975i)T + (13.8 + 18.3i)T^{2} \) |
| 29 | \( 1 + (3.18 - 3.49i)T + (-2.67 - 28.8i)T^{2} \) |
| 31 | \( 1 + (0.283 - 0.0806i)T + (26.3 - 16.3i)T^{2} \) |
| 37 | \( 1 + (3.88 + 10.0i)T + (-27.3 + 24.9i)T^{2} \) |
| 41 | \( 1 + (-5.34 - 5.86i)T + (-3.78 + 40.8i)T^{2} \) |
| 43 | \( 1 + (-3.25 + 8.40i)T + (-31.7 - 28.9i)T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 + (8.50 - 5.26i)T + (23.6 - 47.4i)T^{2} \) |
| 59 | \( 1 + (1.05 - 5.66i)T + (-55.0 - 21.3i)T^{2} \) |
| 61 | \( 1 + (-0.131 - 1.42i)T + (-59.9 + 11.2i)T^{2} \) |
| 67 | \( 1 + (2.40 - 12.8i)T + (-62.4 - 24.2i)T^{2} \) |
| 71 | \( 1 + (0.412 - 0.375i)T + (6.55 - 70.6i)T^{2} \) |
| 73 | \( 1 + (-9.01 - 9.88i)T + (-6.73 + 72.6i)T^{2} \) |
| 79 | \( 1 + (3.99 + 3.64i)T + (7.28 + 78.6i)T^{2} \) |
| 83 | \( 1 + (-0.946 - 5.06i)T + (-77.3 + 29.9i)T^{2} \) |
| 89 | \( 1 + (-7.05 - 9.34i)T + (-24.3 + 85.6i)T^{2} \) |
| 97 | \( 1 + (0.252 - 2.72i)T + (-95.3 - 17.8i)T^{2} \) |
| 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
−10.61760128848827816872154880233, −9.577148329093888620681818486166, −8.902748603070897911292041419758, −8.073998683162300127482157948332, −7.04763238200402342253072186179, −6.06858236534152865547912445343, −5.30558681792426754743582110892, −4.09627629536399158737968051039, −3.16632605444300115519524598251, −2.40632857537854687450863010023,
0.07658730281072703168643428877, 1.87553231383866037935383022260, 3.29317053169897606072425230182, 4.60997048560911476737848975073, 5.04496796971896980636043379826, 6.22099556504813191818343007171, 6.79457316556532765156765514058, 7.86768695849713269604428216784, 9.167924753653819568173097075164, 9.618001053361441033521548788734
