| L(s) = 1 | + (−0.126 + 1.40i)2-s + (0.540 + 0.841i)3-s + (−1.96 − 0.357i)4-s + (−0.902 + 0.412i)5-s + (−1.25 + 0.654i)6-s + (−3.01 + 0.884i)7-s + (0.752 − 2.72i)8-s + (−0.415 + 0.909i)9-s + (−0.465 − 1.32i)10-s + (−2.49 + 2.16i)11-s + (−0.763 − 1.84i)12-s + (0.168 − 0.573i)13-s + (−0.863 − 4.35i)14-s + (−0.834 − 0.536i)15-s + (3.74 + 1.40i)16-s + (1.03 − 7.18i)17-s + ⋯ |
| L(s) = 1 | + (−0.0896 + 0.995i)2-s + (0.312 + 0.485i)3-s + (−0.983 − 0.178i)4-s + (−0.403 + 0.184i)5-s + (−0.511 + 0.267i)6-s + (−1.13 + 0.334i)7-s + (0.266 − 0.963i)8-s + (−0.138 + 0.303i)9-s + (−0.147 − 0.418i)10-s + (−0.753 + 0.652i)11-s + (−0.220 − 0.533i)12-s + (0.0467 − 0.159i)13-s + (−0.230 − 1.16i)14-s + (−0.215 − 0.138i)15-s + (0.936 + 0.351i)16-s + (0.250 − 1.74i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.154 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.154 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.112330 - 0.131307i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.112330 - 0.131307i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.126 - 1.40i)T \) |
| 3 | \( 1 + (-0.540 - 0.841i)T \) |
| 23 | \( 1 + (4.32 + 2.06i)T \) |
| good | 5 | \( 1 + (0.902 - 0.412i)T + (3.27 - 3.77i)T^{2} \) |
| 7 | \( 1 + (3.01 - 0.884i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (2.49 - 2.16i)T + (1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.168 + 0.573i)T + (-10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (-1.03 + 7.18i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-2.99 + 0.429i)T + (18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (3.65 + 0.525i)T + (27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (4.15 + 2.66i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (1.10 + 0.505i)T + (24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (1.87 + 4.11i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-2.11 - 3.29i)T + (-17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 + (-3.26 - 11.1i)T + (-44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (2.63 - 8.98i)T + (-49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (-1.77 + 2.76i)T + (-25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (-11.3 - 9.80i)T + (9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (7.25 - 8.37i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (-0.343 - 2.38i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (-7.19 - 2.11i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (13.8 + 6.33i)T + (54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (-7.20 + 4.63i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (4.27 + 9.35i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−11.36205690590431913926731146159, −9.979640048867277447362166714863, −9.692829154601857723279283856259, −8.814582723039422566712822848517, −7.61057207537794444792378164055, −7.19303546627193031990501022270, −5.88989374108392796811132883966, −5.08223863955027428076898208988, −3.89798443905182237642517013462, −2.83185301078594274949464328366,
0.095585149615088312920064515517, 1.82022464999266913127346397747, 3.32753321037361526569819943939, 3.80391004809188007980950718223, 5.41038376873502231864902848236, 6.46064207803844216490447497410, 7.87013486109455877453909570993, 8.310431515619435967475913887450, 9.470199227165215822106786236631, 10.15177900749595415616490034778
