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
−11.37110008308354668038535459235, −10.59956074990891556705941474705, −9.036869506651318970704853864051, −8.216281296453379748518194295658, −7.42126344996563982577916843835, −6.10766979689044240262078458224, −4.90892997941550771918707336618, −4.49365480257482527608180226765, −2.18475513532811910328119361194, −0.52888160263866085700447182493,
1.41483685647485003493912449335, 3.82432195528885319479914378081, 4.36870334346799375425419599641, 5.71594206519305134506384258123, 6.93606433008364612346485899635, 7.62555503656015854117732150296, 8.742248706554161689382262286731, 10.21439203034714417075792311868, 10.91915613368680149463158243096, 11.53029728273235051245342331724