| L(s) = 1 | + (−0.577 + 0.794i)2-s + (−0.535 + 1.64i)3-s + (0.937 + 2.88i)4-s + (−0.321 + 0.233i)5-s + (−1.00 − 1.37i)6-s + (6.87 − 2.23i)7-s + (−6.57 − 2.13i)8-s + (−2.42 − 1.76i)9-s − 0.390i·10-s + (−0.495 − 10.9i)11-s − 5.25·12-s + (7.14 − 9.83i)13-s + (−2.19 + 6.75i)14-s + (−0.212 − 0.654i)15-s + (−4.32 + 3.14i)16-s + (12.2 + 16.9i)17-s + ⋯ |
| L(s) = 1 | + (−0.288 + 0.397i)2-s + (−0.178 + 0.549i)3-s + (0.234 + 0.721i)4-s + (−0.0643 + 0.0467i)5-s + (−0.166 − 0.229i)6-s + (0.981 − 0.319i)7-s + (−0.821 − 0.266i)8-s + (−0.269 − 0.195i)9-s − 0.0390i·10-s + (−0.0450 − 0.998i)11-s − 0.438·12-s + (0.549 − 0.756i)13-s + (−0.156 + 0.482i)14-s + (−0.0141 − 0.0436i)15-s + (−0.270 + 0.196i)16-s + (0.722 + 0.995i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.332 - 0.943i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.332 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.740072 + 0.523967i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.740072 + 0.523967i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.535 - 1.64i)T \) |
| 11 | \( 1 + (0.495 + 10.9i)T \) |
| good | 2 | \( 1 + (0.577 - 0.794i)T + (-1.23 - 3.80i)T^{2} \) |
| 5 | \( 1 + (0.321 - 0.233i)T + (7.72 - 23.7i)T^{2} \) |
| 7 | \( 1 + (-6.87 + 2.23i)T + (39.6 - 28.8i)T^{2} \) |
| 13 | \( 1 + (-7.14 + 9.83i)T + (-52.2 - 160. i)T^{2} \) |
| 17 | \( 1 + (-12.2 - 16.9i)T + (-89.3 + 274. i)T^{2} \) |
| 19 | \( 1 + (10.5 + 3.43i)T + (292. + 212. i)T^{2} \) |
| 23 | \( 1 + 5.92T + 529T^{2} \) |
| 29 | \( 1 + (-23.7 + 7.71i)T + (680. - 494. i)T^{2} \) |
| 31 | \( 1 + (48.2 + 35.0i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (1.84 + 5.67i)T + (-1.10e3 + 804. i)T^{2} \) |
| 41 | \( 1 + (49.4 + 16.0i)T + (1.35e3 + 988. i)T^{2} \) |
| 43 | \( 1 + 17.6iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (17.1 - 52.6i)T + (-1.78e3 - 1.29e3i)T^{2} \) |
| 53 | \( 1 + (-76.5 - 55.6i)T + (868. + 2.67e3i)T^{2} \) |
| 59 | \( 1 + (7.75 + 23.8i)T + (-2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (-19.6 - 27.0i)T + (-1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 - 94.6T + 4.48e3T^{2} \) |
| 71 | \( 1 + (66.9 - 48.6i)T + (1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (44.2 - 14.3i)T + (4.31e3 - 3.13e3i)T^{2} \) |
| 79 | \( 1 + (-35.4 + 48.7i)T + (-1.92e3 - 5.93e3i)T^{2} \) |
| 83 | \( 1 + (88.5 + 121. i)T + (-2.12e3 + 6.55e3i)T^{2} \) |
| 89 | \( 1 - 134.T + 7.92e3T^{2} \) |
| 97 | \( 1 + (30.4 + 22.1i)T + (2.90e3 + 8.94e3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.83841470540380427412511669712, −15.70946913340282845276138603169, −14.67177978831064240846399255129, −13.11312325713278746898401700556, −11.59953947619937101808307474943, −10.61500599303428760890315780757, −8.704578937130054583066745082693, −7.75997157248207678811117759086, −5.83942512868137111323951083051, −3.69790791199141113395245674101,
1.84109453060867794225412917651, 5.14317502831064095892490996680, 6.83567959305542649579598809256, 8.548629608919472989713084147098, 10.07277072306588281230130397167, 11.41286281683644293628902010613, 12.22084412148722984241309718480, 14.03765389728829146052387489484, 14.90399131987686252622655783461, 16.32224057569001869448386475124
