| L(s) = 1 | + (−1.70 − 1.04i)2-s + (1.81 + 3.56i)4-s + (0.00321 − 0.0182i)5-s + (−0.446 − 0.532i)7-s + (0.620 − 7.97i)8-s + (−0.0245 + 0.0277i)10-s + (−5.71 + 1.00i)11-s + (6.99 − 2.54i)13-s + (0.205 + 1.37i)14-s + (−9.38 + 12.9i)16-s + (4.38 − 7.59i)17-s + (17.1 − 9.88i)19-s + (0.0707 − 0.0216i)20-s + (10.8 + 4.25i)22-s + (−7.15 + 8.52i)23-s + ⋯ |
| L(s) = 1 | + (−0.852 − 0.522i)2-s + (0.454 + 0.890i)4-s + (0.000642 − 0.00364i)5-s + (−0.0638 − 0.0760i)7-s + (0.0775 − 0.996i)8-s + (−0.00245 + 0.00277i)10-s + (−0.519 + 0.0916i)11-s + (0.538 − 0.195i)13-s + (0.0147 + 0.0981i)14-s + (−0.586 + 0.809i)16-s + (0.257 − 0.446i)17-s + (0.901 − 0.520i)19-s + (0.00353 − 0.00108i)20-s + (0.491 + 0.193i)22-s + (−0.310 + 0.370i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.516 + 0.856i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.516 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.923388 - 0.521375i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.923388 - 0.521375i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.70 + 1.04i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-0.00321 + 0.0182i)T + (-23.4 - 8.55i)T^{2} \) |
| 7 | \( 1 + (0.446 + 0.532i)T + (-8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (5.71 - 1.00i)T + (113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (-6.99 + 2.54i)T + (129. - 108. i)T^{2} \) |
| 17 | \( 1 + (-4.38 + 7.59i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-17.1 + 9.88i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (7.15 - 8.52i)T + (-91.8 - 520. i)T^{2} \) |
| 29 | \( 1 + (-21.1 - 7.69i)T + (644. + 540. i)T^{2} \) |
| 31 | \( 1 + (-20.2 + 24.1i)T + (-166. - 946. i)T^{2} \) |
| 37 | \( 1 + (-26.7 + 46.3i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-52.8 + 19.2i)T + (1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (70.5 - 12.4i)T + (1.73e3 - 632. i)T^{2} \) |
| 47 | \( 1 + (43.9 + 52.3i)T + (-383. + 2.17e3i)T^{2} \) |
| 53 | \( 1 - 41.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-50.0 - 8.83i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-28.9 + 24.2i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (14.8 + 40.7i)T + (-3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-49.5 - 28.5i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-14.6 - 25.2i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (46.0 - 126. i)T + (-4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (-49.0 + 134. i)T + (-5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (-29.5 - 51.1i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-6.63 - 37.6i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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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
−11.19031438728346011053853100779, −10.24777563480782235067586338282, −9.486420427740433739003842900010, −8.498776168993472322062059038334, −7.61312326327922641811523010792, −6.67689228857168370043428820592, −5.22006540742164248980167194781, −3.67896467450813686768253293592, −2.51668069118306111099207861984, −0.816448182734942629305442546164,
1.17688109717087597542294545477, 2.88799027535601466823131535522, 4.72859477906863170004005206121, 5.91404453800169673229036470010, 6.74724428830047284128658364882, 7.967268589965736532994802627700, 8.526699944123735494184179440083, 9.716544568595145563304661942998, 10.38096217406419511709610935243, 11.32843531132303340844924565379
