| L(s) = 1 | + (0.322 − 0.143i)2-s + (−0.648 − 1.60i)3-s + (−1.25 + 1.39i)4-s + (−1.80 − 1.31i)5-s + (−0.439 − 0.424i)6-s + (1.84 − 3.20i)7-s + (−0.422 + 1.30i)8-s + (−2.15 + 2.08i)9-s + (−0.771 − 0.164i)10-s + (−5.06 + 2.25i)11-s + (3.05 + 1.11i)12-s + (−3.83 − 1.70i)13-s + (0.136 − 1.29i)14-s + (−0.941 + 3.75i)15-s + (−0.341 − 3.25i)16-s + (0.721 − 2.21i)17-s + ⋯ |
| L(s) = 1 | + (0.227 − 0.101i)2-s + (−0.374 − 0.927i)3-s + (−0.627 + 0.696i)4-s + (−0.808 − 0.588i)5-s + (−0.179 − 0.173i)6-s + (0.698 − 1.20i)7-s + (−0.149 + 0.459i)8-s + (−0.719 + 0.694i)9-s + (−0.244 − 0.0521i)10-s + (−1.52 + 0.680i)11-s + (0.881 + 0.320i)12-s + (−1.06 − 0.473i)13-s + (0.0364 − 0.346i)14-s + (−0.243 + 0.969i)15-s + (−0.0854 − 0.812i)16-s + (0.174 − 0.538i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 + 0.352i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.935 + 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0951166 - 0.522104i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0951166 - 0.522104i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.648 + 1.60i)T \) |
| 5 | \( 1 + (1.80 + 1.31i)T \) |
| good | 2 | \( 1 + (-0.322 + 0.143i)T + (1.33 - 1.48i)T^{2} \) |
| 7 | \( 1 + (-1.84 + 3.20i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (5.06 - 2.25i)T + (7.36 - 8.17i)T^{2} \) |
| 13 | \( 1 + (3.83 + 1.70i)T + (8.69 + 9.66i)T^{2} \) |
| 17 | \( 1 + (-0.721 + 2.21i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-1.49 + 4.61i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-0.0122 + 0.116i)T + (-22.4 - 4.78i)T^{2} \) |
| 29 | \( 1 + (-1.63 - 0.346i)T + (26.4 + 11.7i)T^{2} \) |
| 31 | \( 1 + (-8.52 + 1.81i)T + (28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (-2.35 + 1.71i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (3.36 + 1.49i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (2.06 - 3.58i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.22 - 0.898i)T + (42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (3.62 + 11.1i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (7.00 + 3.12i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + (1.35 - 0.605i)T + (40.8 - 45.3i)T^{2} \) |
| 67 | \( 1 + (-0.659 + 0.140i)T + (61.2 - 27.2i)T^{2} \) |
| 71 | \( 1 + (-1.99 - 6.12i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (9.08 + 6.60i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (8.44 + 1.79i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (-2.71 - 3.01i)T + (-8.67 + 82.5i)T^{2} \) |
| 89 | \( 1 + (7.85 + 5.70i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-9.42 - 2.00i)T + (88.6 + 39.4i)T^{2} \) |
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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
−11.99374011686134385460262530769, −11.20418142426251784262791590801, −9.996621617680947012532399603152, −8.405437468235595729504010667664, −7.61630053462085937507203701430, −7.32410741647760721226755469233, −4.99359079502404541415369923837, −4.69502518197191149903653301866, −2.80812803497728479820969572312, −0.42393664692027002020834919834,
2.90571734836907751046341186987, 4.44243692703401266140886665650, 5.28240043100334097915110853422, 6.12448696232616586910573153949, 7.943898723086873068045591833293, 8.757022730944191515155812440196, 10.03078418107592288710082566805, 10.59841205735979848639656898664, 11.69645839624632501178572263059, 12.36824548013050086347547674047
