| L(s) = 1 | + (−2.92 − 0.515i)5-s + (0.715 − 0.600i)7-s + (−5.89 + 1.04i)11-s + (−9.53 + 3.47i)13-s + (−6.81 − 3.93i)17-s + (−2.29 − 3.97i)19-s + (−22.9 + 27.3i)23-s + (−15.2 − 5.53i)25-s + (−3.20 + 8.80i)29-s + (−41.5 − 34.8i)31-s + (−2.40 + 1.38i)35-s + (−9.21 + 15.9i)37-s + (3.70 + 10.1i)41-s + (−9.70 − 55.0i)43-s + (46.8 + 55.8i)47-s + ⋯ |
| L(s) = 1 | + (−0.584 − 0.103i)5-s + (0.102 − 0.0858i)7-s + (−0.536 + 0.0945i)11-s + (−0.733 + 0.266i)13-s + (−0.400 − 0.231i)17-s + (−0.120 − 0.208i)19-s + (−0.996 + 1.18i)23-s + (−0.608 − 0.221i)25-s + (−0.110 + 0.303i)29-s + (−1.34 − 1.12i)31-s + (−0.0686 + 0.0396i)35-s + (−0.249 + 0.431i)37-s + (0.0903 + 0.248i)41-s + (−0.225 − 1.28i)43-s + (0.997 + 1.18i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0836i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.996 - 0.0836i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.00127522 + 0.0304300i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00127522 + 0.0304300i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (2.92 + 0.515i)T + (23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (-0.715 + 0.600i)T + (8.50 - 48.2i)T^{2} \) |
| 11 | \( 1 + (5.89 - 1.04i)T + (113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (9.53 - 3.47i)T + (129. - 108. i)T^{2} \) |
| 17 | \( 1 + (6.81 + 3.93i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (2.29 + 3.97i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (22.9 - 27.3i)T + (-91.8 - 520. i)T^{2} \) |
| 29 | \( 1 + (3.20 - 8.80i)T + (-644. - 540. i)T^{2} \) |
| 31 | \( 1 + (41.5 + 34.8i)T + (166. + 946. i)T^{2} \) |
| 37 | \( 1 + (9.21 - 15.9i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-3.70 - 10.1i)T + (-1.28e3 + 1.08e3i)T^{2} \) |
| 43 | \( 1 + (9.70 + 55.0i)T + (-1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (-46.8 - 55.8i)T + (-383. + 2.17e3i)T^{2} \) |
| 53 | \( 1 + 44.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (81.5 + 14.3i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-47.6 + 40.0i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (31.9 - 11.6i)T + (3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (60.7 + 35.0i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-68.4 - 118. i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-41.4 - 15.0i)T + (4.78e3 + 4.01e3i)T^{2} \) |
| 83 | \( 1 + (-31.4 + 86.3i)T + (-5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (-86.0 + 49.6i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (4.81 + 27.3i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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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
−11.79020706037754520221529386319, −11.05131920475923609972366276493, −9.950290274452650117754255394163, −9.096041356665007123102620787241, −7.86178731486273608847266996471, −7.31388032765016781125072839169, −5.91825526340665086652107191411, −4.77061634932074214393182021568, −3.68592265764612284624812389890, −2.12226676294836783568671824197,
0.01334333292122971524020124094, 2.22022553323906335058350424315, 3.65822948340043852832783828395, 4.82562333717299500937708007035, 5.98513978300333841355600328967, 7.23387996976193781463354685983, 8.021348994816016024891108052039, 8.975695634777488660443485394312, 10.18585554906922556667784815977, 10.86889148054421317843746483712
