L(s) = 1 | + (0.965 − 0.258i)2-s + (−0.125 + 1.72i)3-s + (0.866 − 0.499i)4-s + (−1.36 + 1.77i)5-s + (0.325 + 1.70i)6-s + (2.62 + 0.301i)7-s + (0.707 − 0.707i)8-s + (−2.96 − 0.433i)9-s + (−0.854 + 2.06i)10-s + (−1.49 + 0.860i)11-s + (0.755 + 1.55i)12-s + (1.71 + 1.71i)13-s + (2.61 − 0.389i)14-s + (−2.89 − 2.57i)15-s + (0.500 − 0.866i)16-s + (0.641 − 2.39i)17-s + ⋯ |
L(s) = 1 | + (0.683 − 0.183i)2-s + (−0.0724 + 0.997i)3-s + (0.433 − 0.249i)4-s + (−0.608 + 0.793i)5-s + (0.133 + 0.694i)6-s + (0.993 + 0.113i)7-s + (0.249 − 0.249i)8-s + (−0.989 − 0.144i)9-s + (−0.270 + 0.653i)10-s + (−0.449 + 0.259i)11-s + (0.217 + 0.449i)12-s + (0.474 + 0.474i)13-s + (0.699 − 0.104i)14-s + (−0.747 − 0.664i)15-s + (0.125 − 0.216i)16-s + (0.155 − 0.580i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.518 - 0.854i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.518 - 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.43521 + 0.807981i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.43521 + 0.807981i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.965 + 0.258i)T \) |
| 3 | \( 1 + (0.125 - 1.72i)T \) |
| 5 | \( 1 + (1.36 - 1.77i)T \) |
| 7 | \( 1 + (-2.62 - 0.301i)T \) |
good | 11 | \( 1 + (1.49 - 0.860i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.71 - 1.71i)T + 13iT^{2} \) |
| 17 | \( 1 + (-0.641 + 2.39i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-3.74 - 2.16i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.97 + 7.36i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 - 3.49T + 29T^{2} \) |
| 31 | \( 1 + (2.74 + 4.75i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.29 + 4.83i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 - 5.76iT - 41T^{2} \) |
| 43 | \( 1 + (1.33 + 1.33i)T + 43iT^{2} \) |
| 47 | \( 1 + (5.84 - 1.56i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-13.1 - 3.51i)T + (45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (5.45 + 9.44i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.22 - 7.31i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (13.2 + 3.54i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 10.9iT - 71T^{2} \) |
| 73 | \( 1 + (0.849 - 3.17i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-2.97 - 1.71i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.66 - 1.66i)T - 83iT^{2} \) |
| 89 | \( 1 + (1.63 - 2.83i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-11.1 + 11.1i)T - 97iT^{2} \) |
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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
−12.18691982663066711358414952497, −11.49594203489087125542716423374, −10.79051804519857120057063624172, −9.954935031415249524765213745981, −8.528853280854775284123973377932, −7.46357353375813939655322346380, −6.05743321469169754874733603549, −4.85146566249993680708167587888, −3.96140404889282519230610740676, −2.64079270328394043939350602852,
1.44537296441089989104413979561, 3.39288137272412819646429160347, 4.97219317292033641644406832565, 5.74862717951624701158415132001, 7.29226655866780954432649157271, 7.970030913823788517626913851375, 8.770731868309333676452526365375, 10.69982815765222589096284419534, 11.68266895089616933634012143307, 12.15998389401015038228320498528