| L(s) = 1 | + (−0.298 − 1.38i)2-s + (−0.512 − 1.65i)3-s + (−1.82 + 0.824i)4-s + (1.00 − 0.705i)5-s + (−2.13 + 1.20i)6-s + (1.76 − 1.47i)7-s + (1.68 + 2.27i)8-s + (−2.47 + 1.69i)9-s + (−1.27 − 1.18i)10-s + (−3.48 − 2.43i)11-s + (2.29 + 2.59i)12-s + (−4.80 − 2.24i)13-s + (−2.56 − 1.99i)14-s + (−1.68 − 1.30i)15-s + (2.64 − 3.00i)16-s + (−4.87 − 2.81i)17-s + ⋯ |
| L(s) = 1 | + (−0.210 − 0.977i)2-s + (−0.295 − 0.955i)3-s + (−0.911 + 0.412i)4-s + (0.450 − 0.315i)5-s + (−0.871 + 0.490i)6-s + (0.665 − 0.558i)7-s + (0.594 + 0.803i)8-s + (−0.825 + 0.564i)9-s + (−0.403 − 0.374i)10-s + (−1.04 − 0.734i)11-s + (0.663 + 0.748i)12-s + (−1.33 − 0.621i)13-s + (−0.686 − 0.532i)14-s + (−0.434 − 0.337i)15-s + (0.660 − 0.750i)16-s + (−1.18 − 0.682i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.829 - 0.557i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.829 - 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.225267 + 0.739053i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.225267 + 0.739053i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.298 + 1.38i)T \) |
| 3 | \( 1 + (0.512 + 1.65i)T \) |
| good | 5 | \( 1 + (-1.00 + 0.705i)T + (1.71 - 4.69i)T^{2} \) |
| 7 | \( 1 + (-1.76 + 1.47i)T + (1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (3.48 + 2.43i)T + (3.76 + 10.3i)T^{2} \) |
| 13 | \( 1 + (4.80 + 2.24i)T + (8.35 + 9.95i)T^{2} \) |
| 17 | \( 1 + (4.87 + 2.81i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-7.38 - 1.97i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (-0.853 + 1.01i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-1.09 + 0.510i)T + (18.6 - 22.2i)T^{2} \) |
| 31 | \( 1 + (4.93 - 5.88i)T + (-5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (0.993 + 3.70i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-3.24 + 1.18i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.83 + 2.62i)T + (-14.7 - 40.4i)T^{2} \) |
| 47 | \( 1 + (-5.41 + 4.54i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (5.24 + 5.24i)T + 53iT^{2} \) |
| 59 | \( 1 + (7.92 + 11.3i)T + (-20.1 + 55.4i)T^{2} \) |
| 61 | \( 1 + (-6.93 - 0.607i)T + (60.0 + 10.5i)T^{2} \) |
| 67 | \( 1 + (-4.67 + 10.0i)T + (-43.0 - 51.3i)T^{2} \) |
| 71 | \( 1 + (-8.91 - 5.14i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (3.53 - 2.03i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.03 + 2.84i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-0.259 - 0.556i)T + (-53.3 + 63.5i)T^{2} \) |
| 89 | \( 1 + (5.58 + 9.67i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.91 + 10.8i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−10.88062653281611038937702320033, −9.871758332901670020039090447142, −8.851290904560781735455941244535, −7.80309140061940854772468040015, −7.28699033969877128759005907410, −5.40703709401877506949931651066, −5.00932961959220600889913469199, −3.09982002005700317454755624476, −1.98231966370798416334179200119, −0.53263056948837720457401556370,
2.45578635128741705843340261609, 4.39173904885853098175309825539, 5.06031021544239646445950161493, 5.86723021615228377516905742360, 7.07277507656373101987876784190, 7.980271980714790691167001610898, 9.177260226893127954964643951384, 9.668683288276921460841511547614, 10.52088338913939351771339540016, 11.49206496823016723652487319440
