| L(s) = 1 | + (0.959 − 0.281i)2-s + (−0.654 − 0.755i)3-s + (0.841 − 0.540i)4-s + (−0.592 − 4.11i)5-s + (−0.841 − 0.540i)6-s + (−1.22 + 2.67i)7-s + (0.654 − 0.755i)8-s + (−0.142 + 0.989i)9-s + (−1.72 − 3.78i)10-s + (3.15 + 0.925i)11-s + (−0.959 − 0.281i)12-s + (−0.0583 − 0.127i)13-s + (−0.418 + 2.91i)14-s + (−2.72 + 3.14i)15-s + (0.415 − 0.909i)16-s + (5.26 + 3.38i)17-s + ⋯ |
| L(s) = 1 | + (0.678 − 0.199i)2-s + (−0.378 − 0.436i)3-s + (0.420 − 0.270i)4-s + (−0.264 − 1.84i)5-s + (−0.343 − 0.220i)6-s + (−0.462 + 1.01i)7-s + (0.231 − 0.267i)8-s + (−0.0474 + 0.329i)9-s + (−0.546 − 1.19i)10-s + (0.950 + 0.278i)11-s + (−0.276 − 0.0813i)12-s + (−0.0161 − 0.0354i)13-s + (−0.111 + 0.778i)14-s + (−0.703 + 0.812i)15-s + (0.103 − 0.227i)16-s + (1.27 + 0.819i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 + 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.309 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.09472 - 0.795240i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.09472 - 0.795240i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.959 + 0.281i)T \) |
| 3 | \( 1 + (0.654 + 0.755i)T \) |
| 23 | \( 1 + (0.435 - 4.77i)T \) |
| good | 5 | \( 1 + (0.592 + 4.11i)T + (-4.79 + 1.40i)T^{2} \) |
| 7 | \( 1 + (1.22 - 2.67i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-3.15 - 0.925i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (0.0583 + 0.127i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (-5.26 - 3.38i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (-1.99 + 1.28i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (8.36 + 5.37i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (1.49 - 1.72i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (0.883 - 6.14i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (0.918 + 6.38i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-0.839 - 0.968i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 - 2.84T + 47T^{2} \) |
| 53 | \( 1 + (-1.80 + 3.95i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (-0.0304 - 0.0666i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (1.25 - 1.45i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (4.15 - 1.22i)T + (56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (9.39 - 2.75i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-4.61 + 2.96i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (-3.99 - 8.75i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-1.56 + 10.8i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (8.58 + 9.90i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-1.73 - 12.0i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−12.77027375225601702398325681517, −12.10876053142390089085771136595, −11.64418595227226843012181371497, −9.736576971241219826548367849108, −8.877522117921358981806342637681, −7.62717329751451650537649609311, −5.92851332082535909301523248232, −5.27514733305033244201626182799, −3.84370645904478803440149266113, −1.53153703662212262234651981373,
3.19767855795492488134644991848, 3.95202562856841064321686921745, 5.81574568205274835487273680288, 6.87284524041850998733874796876, 7.49655086624846663240249887644, 9.620855568317676055142841020431, 10.59113277887344739793923021781, 11.26359981827136796732957627550, 12.24585734051528563426235481785, 13.76524229950870614764602314752
