| L(s) = 1 | + (2.91 − 0.722i)3-s + (−2.43 + 1.81i)5-s + (−7.41 + 4.87i)7-s + (7.95 − 4.20i)9-s + (−2.42 + 20.7i)11-s + (0.0230 + 0.0770i)13-s + (−5.79 + 7.04i)15-s + (10.3 + 1.82i)17-s + (4.67 + 26.5i)19-s + (−18.0 + 19.5i)21-s + (3.75 − 5.71i)23-s + (−4.51 + 15.0i)25-s + (20.1 − 17.9i)27-s + (6.47 − 6.10i)29-s + (2.04 − 35.0i)31-s + ⋯ |
| L(s) = 1 | + (0.970 − 0.240i)3-s + (−0.487 + 0.363i)5-s + (−1.05 + 0.697i)7-s + (0.884 − 0.467i)9-s + (−0.220 + 1.88i)11-s + (0.00177 + 0.00592i)13-s + (−0.386 + 0.469i)15-s + (0.607 + 0.107i)17-s + (0.246 + 1.39i)19-s + (−0.860 + 0.931i)21-s + (0.163 − 0.248i)23-s + (−0.180 + 0.603i)25-s + (0.745 − 0.666i)27-s + (0.223 − 0.210i)29-s + (0.0659 − 1.13i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.113 - 0.993i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.113 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.22842 + 1.09613i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.22842 + 1.09613i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.91 + 0.722i)T \) |
| good | 5 | \( 1 + (2.43 - 1.81i)T + (7.17 - 23.9i)T^{2} \) |
| 7 | \( 1 + (7.41 - 4.87i)T + (19.4 - 44.9i)T^{2} \) |
| 11 | \( 1 + (2.42 - 20.7i)T + (-117. - 27.9i)T^{2} \) |
| 13 | \( 1 + (-0.0230 - 0.0770i)T + (-141. + 92.8i)T^{2} \) |
| 17 | \( 1 + (-10.3 - 1.82i)T + (271. + 98.8i)T^{2} \) |
| 19 | \( 1 + (-4.67 - 26.5i)T + (-339. + 123. i)T^{2} \) |
| 23 | \( 1 + (-3.75 + 5.71i)T + (-209. - 485. i)T^{2} \) |
| 29 | \( 1 + (-6.47 + 6.10i)T + (48.8 - 839. i)T^{2} \) |
| 31 | \( 1 + (-2.04 + 35.0i)T + (-954. - 111. i)T^{2} \) |
| 37 | \( 1 + (44.6 - 16.2i)T + (1.04e3 - 879. i)T^{2} \) |
| 41 | \( 1 + (-15.1 + 63.8i)T + (-1.50e3 - 754. i)T^{2} \) |
| 43 | \( 1 + (-2.90 - 6.72i)T + (-1.26e3 + 1.34e3i)T^{2} \) |
| 47 | \( 1 + (-3.18 + 0.185i)T + (2.19e3 - 256. i)T^{2} \) |
| 53 | \( 1 + (-64.8 - 37.4i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-8.93 - 76.4i)T + (-3.38e3 + 802. i)T^{2} \) |
| 61 | \( 1 + (72.0 + 36.1i)T + (2.22e3 + 2.98e3i)T^{2} \) |
| 67 | \( 1 + (42.7 - 45.2i)T + (-261. - 4.48e3i)T^{2} \) |
| 71 | \( 1 + (-14.0 + 16.7i)T + (-875. - 4.96e3i)T^{2} \) |
| 73 | \( 1 + (82.0 - 68.8i)T + (925. - 5.24e3i)T^{2} \) |
| 79 | \( 1 + (-107. + 25.4i)T + (5.57e3 - 2.80e3i)T^{2} \) |
| 83 | \( 1 + (8.85 + 37.3i)T + (-6.15e3 + 3.09e3i)T^{2} \) |
| 89 | \( 1 + (-34.5 - 41.2i)T + (-1.37e3 + 7.80e3i)T^{2} \) |
| 97 | \( 1 + (-59.3 + 79.7i)T + (-2.69e3 - 9.01e3i)T^{2} \) |
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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
−12.13141490004267359972843533371, −10.28714995531005938580720662627, −9.765812220068790820504850794966, −8.854215707498526666788093672269, −7.63420433089870642219996251912, −7.12202080365455719629296080365, −5.83786287256306924491860955446, −4.19872853391570795890333928134, −3.16821625183915814575732826994, −1.98831572614650736842282577873,
0.69761262496837766671100067358, 3.02075935575189567768590782378, 3.60434053643011875770423734164, 4.96619857501614400132872244524, 6.45241003965733329880843691116, 7.49933640438701745987464553200, 8.474621025931288775943423924860, 9.144471242745722524603625608289, 10.20457758659768660628040297451, 10.99760934313053334983740941881
