| L(s) = 1 | + (−2.85 + 0.915i)3-s + (−4.55 + 3.39i)5-s + (9.74 − 6.40i)7-s + (7.32 − 5.23i)9-s + (0.713 − 6.10i)11-s + (4.08 + 13.6i)13-s + (9.91 − 13.8i)15-s + (−21.2 − 3.75i)17-s + (6.10 + 34.6i)19-s + (−21.9 + 27.2i)21-s + (−11.5 + 17.4i)23-s + (2.09 − 7.00i)25-s + (−16.1 + 21.6i)27-s + (4.94 − 4.66i)29-s + (−2.02 + 34.7i)31-s + ⋯ |
| L(s) = 1 | + (−0.952 + 0.305i)3-s + (−0.911 + 0.678i)5-s + (1.39 − 0.915i)7-s + (0.813 − 0.581i)9-s + (0.0648 − 0.554i)11-s + (0.313 + 1.04i)13-s + (0.661 − 0.924i)15-s + (−1.25 − 0.220i)17-s + (0.321 + 1.82i)19-s + (−1.04 + 1.29i)21-s + (−0.500 + 0.760i)23-s + (0.0839 − 0.280i)25-s + (−0.597 + 0.802i)27-s + (0.170 − 0.160i)29-s + (−0.0653 + 1.12i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.154 - 0.988i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.154 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.586528 + 0.685323i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.586528 + 0.685323i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.85 - 0.915i)T \) |
| good | 5 | \( 1 + (4.55 - 3.39i)T + (7.17 - 23.9i)T^{2} \) |
| 7 | \( 1 + (-9.74 + 6.40i)T + (19.4 - 44.9i)T^{2} \) |
| 11 | \( 1 + (-0.713 + 6.10i)T + (-117. - 27.9i)T^{2} \) |
| 13 | \( 1 + (-4.08 - 13.6i)T + (-141. + 92.8i)T^{2} \) |
| 17 | \( 1 + (21.2 + 3.75i)T + (271. + 98.8i)T^{2} \) |
| 19 | \( 1 + (-6.10 - 34.6i)T + (-339. + 123. i)T^{2} \) |
| 23 | \( 1 + (11.5 - 17.4i)T + (-209. - 485. i)T^{2} \) |
| 29 | \( 1 + (-4.94 + 4.66i)T + (48.8 - 839. i)T^{2} \) |
| 31 | \( 1 + (2.02 - 34.7i)T + (-954. - 111. i)T^{2} \) |
| 37 | \( 1 + (-14.3 + 5.21i)T + (1.04e3 - 879. i)T^{2} \) |
| 41 | \( 1 + (0.792 - 3.34i)T + (-1.50e3 - 754. i)T^{2} \) |
| 43 | \( 1 + (5.34 + 12.3i)T + (-1.26e3 + 1.34e3i)T^{2} \) |
| 47 | \( 1 + (-29.6 + 1.72i)T + (2.19e3 - 256. i)T^{2} \) |
| 53 | \( 1 + (-58.2 - 33.6i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-5.87 - 50.2i)T + (-3.38e3 + 802. i)T^{2} \) |
| 61 | \( 1 + (19.7 + 9.92i)T + (2.22e3 + 2.98e3i)T^{2} \) |
| 67 | \( 1 + (72.8 - 77.1i)T + (-261. - 4.48e3i)T^{2} \) |
| 71 | \( 1 + (63.7 - 76.0i)T + (-875. - 4.96e3i)T^{2} \) |
| 73 | \( 1 + (40.8 - 34.2i)T + (925. - 5.24e3i)T^{2} \) |
| 79 | \( 1 + (-70.7 + 16.7i)T + (5.57e3 - 2.80e3i)T^{2} \) |
| 83 | \( 1 + (-20.3 - 85.6i)T + (-6.15e3 + 3.09e3i)T^{2} \) |
| 89 | \( 1 + (17.1 + 20.4i)T + (-1.37e3 + 7.80e3i)T^{2} \) |
| 97 | \( 1 + (-25.5 + 34.3i)T + (-2.69e3 - 9.01e3i)T^{2} \) |
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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
−11.53029728273235051245342331724, −10.91915613368680149463158243096, −10.21439203034714417075792311868, −8.742248706554161689382262286731, −7.62555503656015854117732150296, −6.93606433008364612346485899635, −5.71594206519305134506384258123, −4.36870334346799375425419599641, −3.82432195528885319479914378081, −1.41483685647485003493912449335,
0.52888160263866085700447182493, 2.18475513532811910328119361194, 4.49365480257482527608180226765, 4.90892997941550771918707336618, 6.10766979689044240262078458224, 7.42126344996563982577916843835, 8.216281296453379748518194295658, 9.036869506651318970704853864051, 10.59956074990891556705941474705, 11.37110008308354668038535459235
