| L(s) = 1 | + (−0.989 − 0.142i)2-s + (0.909 − 0.415i)3-s + (0.959 + 0.281i)4-s + (−1.21 − 1.87i)5-s + (−0.959 + 0.281i)6-s + (0.707 − 1.10i)7-s + (−0.909 − 0.415i)8-s + (0.654 − 0.755i)9-s + (0.933 + 2.03i)10-s + (0.267 + 1.85i)11-s + (0.989 − 0.142i)12-s + (−2.09 − 3.25i)13-s + (−0.856 + 0.989i)14-s + (−1.88 − 1.20i)15-s + (0.841 + 0.540i)16-s + (−1.13 − 3.87i)17-s + ⋯ |
| L(s) = 1 | + (−0.699 − 0.100i)2-s + (0.525 − 0.239i)3-s + (0.479 + 0.140i)4-s + (−0.542 − 0.840i)5-s + (−0.391 + 0.115i)6-s + (0.267 − 0.416i)7-s + (−0.321 − 0.146i)8-s + (0.218 − 0.251i)9-s + (0.295 + 0.642i)10-s + (0.0805 + 0.560i)11-s + (0.285 − 0.0410i)12-s + (−0.580 − 0.903i)13-s + (−0.229 + 0.264i)14-s + (−0.486 − 0.311i)15-s + (0.210 + 0.135i)16-s + (−0.276 − 0.940i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.607 + 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.607 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.417520 - 0.845568i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.417520 - 0.845568i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.989 + 0.142i)T \) |
| 3 | \( 1 + (-0.909 + 0.415i)T \) |
| 5 | \( 1 + (1.21 + 1.87i)T \) |
| 23 | \( 1 + (3.69 + 3.06i)T \) |
| good | 7 | \( 1 + (-0.707 + 1.10i)T + (-2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (-0.267 - 1.85i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (2.09 + 3.25i)T + (-5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (1.13 + 3.87i)T + (-14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (-4.77 - 1.40i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (6.75 - 1.98i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-2.27 + 4.97i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (3.36 + 2.91i)T + (5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (4.58 + 5.29i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-2.17 + 0.995i)T + (28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 - 5.44iT - 47T^{2} \) |
| 53 | \( 1 + (-4.25 + 6.62i)T + (-22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (-0.186 + 0.119i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-1.20 + 2.62i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (4.34 + 0.625i)T + (64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (-1.57 + 10.9i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-0.479 + 1.63i)T + (-61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (5.25 - 3.37i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-1.99 - 1.72i)T + (11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-1.12 - 2.45i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (-6.97 + 6.04i)T + (13.8 - 96.0i)T^{2} \) |
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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
−9.895451191886521486398248260862, −9.323673756990207061747477187849, −8.385534253381609384322309819601, −7.58511544582611310651442881344, −7.22126845070468951489225218804, −5.61722067891247240087892826492, −4.54725063316651364306911340075, −3.39708684628656539694841237002, −2.01664906049199929704233728917, −0.57633496738848287181443196489,
1.87768603072425921647392404722, 3.06179141968187491181699295191, 4.07165934978059422806956389132, 5.49842158692905731089142552602, 6.63969786065989379046991078960, 7.43280841176501602262456754263, 8.231594856408954409899590938104, 8.995024147413999786331353189759, 9.870357714036008168924345434714, 10.61027173855121380507963025691
