| L(s) = 1 | + (0.951 − 0.309i)2-s + (0.928 + 1.46i)3-s + (0.809 − 0.587i)4-s + (−0.866 + 0.5i)5-s + (1.33 + 1.10i)6-s + (0.0557 + 0.530i)7-s + (0.587 − 0.809i)8-s + (−1.27 + 2.71i)9-s + (−0.669 + 0.743i)10-s + (3.26 + 1.45i)11-s + (1.61 + 0.637i)12-s + (−0.192 − 0.903i)13-s + (0.217 + 0.487i)14-s + (−1.53 − 0.802i)15-s + (0.309 − 0.951i)16-s + (−4.44 + 1.98i)17-s + ⋯ |
| L(s) = 1 | + (0.672 − 0.218i)2-s + (0.535 + 0.844i)3-s + (0.404 − 0.293i)4-s + (−0.387 + 0.223i)5-s + (0.544 + 0.450i)6-s + (0.0210 + 0.200i)7-s + (0.207 − 0.286i)8-s + (−0.425 + 0.904i)9-s + (−0.211 + 0.235i)10-s + (0.985 + 0.438i)11-s + (0.464 + 0.184i)12-s + (−0.0532 − 0.250i)13-s + (0.0580 + 0.130i)14-s + (−0.396 − 0.207i)15-s + (0.0772 − 0.237i)16-s + (−1.07 + 0.480i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.450 - 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.450 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.28082 + 1.40456i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.28082 + 1.40456i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.951 + 0.309i)T \) |
| 3 | \( 1 + (-0.928 - 1.46i)T \) |
| 5 | \( 1 + (0.866 - 0.5i)T \) |
| 31 | \( 1 + (2.08 - 5.16i)T \) |
| good | 7 | \( 1 + (-0.0557 - 0.530i)T + (-6.84 + 1.45i)T^{2} \) |
| 11 | \( 1 + (-3.26 - 1.45i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (0.192 + 0.903i)T + (-11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (4.44 - 1.98i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (-6.98 - 1.48i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (-2.96 - 2.15i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-1.61 - 4.97i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (6.62 + 3.82i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (7.30 + 6.57i)T + (4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (-0.884 + 4.16i)T + (-39.2 - 17.4i)T^{2} \) |
| 47 | \( 1 + (-10.3 - 3.36i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (0.214 - 2.04i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (5.93 - 5.34i)T + (6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + 2.90iT - 61T^{2} \) |
| 67 | \( 1 + (7.79 + 13.5i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-10.3 - 1.09i)T + (69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (-6.67 + 14.9i)T + (-48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (3.95 + 8.88i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-2.52 + 2.80i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (1.09 - 0.796i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-6.70 + 4.87i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.49626225173754120621674805285, −9.232361527988811802965353022599, −8.891047162953022324201063549336, −7.57048287648801562503436290375, −6.85428063251730094793882440817, −5.56562721845599724802822144155, −4.79969646215847233613318991667, −3.77524213904218826094168546092, −3.18770896679691131845350192932, −1.81992689684686171639152629111,
1.03888295218491055372741995032, 2.52687393824338353519557850278, 3.54561001211662821346286452972, 4.47390998261964809447035459459, 5.65140029803319732157543903412, 6.73846855989907994915247624021, 7.13045556679300146561180101381, 8.150957710252106917604616348464, 8.902030574061687565978474989908, 9.677032362116709236946178313345
