| L(s) = 1 | + (0.359 + 1.36i)2-s + (1.00 + 1.20i)3-s + (−1.74 + 0.983i)4-s + (−1.95 + 1.08i)5-s + (−1.28 + 1.81i)6-s + (−0.0550 + 0.0952i)7-s + (−1.97 − 2.02i)8-s + (0.0930 − 0.527i)9-s + (−2.18 − 2.28i)10-s + (−5.05 + 2.91i)11-s + (−2.94 − 1.10i)12-s + (2.48 + 2.08i)13-s + (−0.150 − 0.0409i)14-s + (−3.27 − 1.25i)15-s + (2.06 − 3.42i)16-s + (−0.624 + 0.110i)17-s + ⋯ |
| L(s) = 1 | + (0.254 + 0.967i)2-s + (0.582 + 0.694i)3-s + (−0.870 + 0.491i)4-s + (−0.874 + 0.485i)5-s + (−0.523 + 0.739i)6-s + (−0.0207 + 0.0360i)7-s + (−0.696 − 0.717i)8-s + (0.0310 − 0.175i)9-s + (−0.691 − 0.722i)10-s + (−1.52 + 0.879i)11-s + (−0.848 − 0.318i)12-s + (0.689 + 0.578i)13-s + (−0.0401 − 0.0109i)14-s + (−0.846 − 0.324i)15-s + (0.516 − 0.856i)16-s + (−0.151 + 0.0267i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.205i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 + 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.117317 - 1.13224i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.117317 - 1.13224i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.359 - 1.36i)T \) |
| 5 | \( 1 + (1.95 - 1.08i)T \) |
| 19 | \( 1 + (-2.31 - 3.69i)T \) |
| good | 3 | \( 1 + (-1.00 - 1.20i)T + (-0.520 + 2.95i)T^{2} \) |
| 7 | \( 1 + (0.0550 - 0.0952i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (5.05 - 2.91i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.48 - 2.08i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.624 - 0.110i)T + (15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (3.43 - 1.24i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-5.65 - 0.996i)T + (27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (3.25 - 5.64i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 0.716T + 37T^{2} \) |
| 41 | \( 1 + (4.84 + 5.77i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-8.35 - 3.04i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (2.13 - 12.1i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (-6.30 + 2.29i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (2.21 + 12.5i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (11.2 - 4.10i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-7.89 - 1.39i)T + (62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (4.58 + 1.66i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-0.0197 - 0.0235i)T + (-12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (1.17 - 0.985i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (2.11 - 3.66i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-7.35 + 8.76i)T + (-15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-0.749 - 4.25i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−12.09410694062216420188703155069, −10.69104936561977294796649555369, −9.889942812005240844299353413581, −8.881019231697485981620837230173, −8.006152842502706007138849347646, −7.31079424048882467887943298711, −6.18948304883902710974880198181, −4.84051666232483890554021349626, −3.97401881054691892224115409838, −3.01555605905064080073647464899,
0.66682962939314141213201214633, 2.44553225574323797826839248564, 3.41298918214325075432787196462, 4.74299546485911765771790253451, 5.73439449234797729993086617173, 7.46571016772613465187557686231, 8.283196271481923852262876799242, 8.764071411848391068052856697021, 10.24751340019100452244710957297, 10.96007646121345340046220389563
