| L(s) = 1 | + (−0.654 + 1.25i)2-s + (−0.684 + 1.59i)3-s + (−1.14 − 1.64i)4-s + (1.15 − 3.17i)5-s + (−1.54 − 1.89i)6-s + (0.593 − 3.36i)7-s + (2.80 − 0.361i)8-s + (−2.06 − 2.17i)9-s + (3.22 + 3.52i)10-s + (0.197 + 0.541i)11-s + (3.39 − 0.696i)12-s + (−4.30 − 5.12i)13-s + (3.83 + 2.94i)14-s + (4.25 + 4.00i)15-s + (−1.38 + 3.75i)16-s + (1.15 + 2.00i)17-s + ⋯ |
| L(s) = 1 | + (−0.462 + 0.886i)2-s + (−0.395 + 0.918i)3-s + (−0.572 − 0.820i)4-s + (0.516 − 1.41i)5-s + (−0.631 − 0.775i)6-s + (0.224 − 1.27i)7-s + (0.991 − 0.127i)8-s + (−0.687 − 0.726i)9-s + (1.01 + 1.11i)10-s + (0.0594 + 0.163i)11-s + (0.979 − 0.201i)12-s + (−1.19 − 1.42i)13-s + (1.02 + 0.787i)14-s + (1.09 + 1.03i)15-s + (−0.345 + 0.938i)16-s + (0.281 + 0.486i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.273i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.961 + 0.273i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.770286 - 0.107227i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.770286 - 0.107227i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.654 - 1.25i)T \) |
| 3 | \( 1 + (0.684 - 1.59i)T \) |
| good | 5 | \( 1 + (-1.15 + 3.17i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.593 + 3.36i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (-0.197 - 0.541i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (4.30 + 5.12i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.15 - 2.00i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.353 - 0.203i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.15 - 6.53i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-2.24 + 2.67i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.382 + 2.17i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-1.05 + 0.607i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.09 + 4.27i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (0.442 + 1.21i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (0.547 - 3.10i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 9.35iT - 53T^{2} \) |
| 59 | \( 1 + (-3.00 + 8.25i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-4.67 - 0.824i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-9.67 - 11.5i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (3.92 + 6.79i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.641 + 1.11i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.84 - 3.22i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (6.67 - 7.95i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (2.86 - 4.96i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-10.5 + 3.84i)T + (74.3 - 62.3i)T^{2} \) |
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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
−12.42814906727304282498928186058, −10.90344905385131360293633068796, −9.957374271933503449917322129372, −9.533646158670398174041443362288, −8.304182583582770710824627365953, −7.39828516857832762964555794401, −5.75228182097420467039988558949, −5.08974455815579743832528143148, −4.12818736862369901120205197287, −0.831293000029621254155007556421,
2.13503576253633138088487207185, 2.76710455466584991353163255572, 5.00099739935268555775620192452, 6.46216846019805407425942509325, 7.24109571265999057602499586734, 8.534322446945615965032702031251, 9.553930805611998587028713603532, 10.60621140407231799677650408357, 11.57451136901310408993776807758, 12.01090548644887681326492622476
