| L(s) = 1 | + (−1.28 + 0.593i)2-s + (−1.30 − 1.13i)3-s + (1.29 − 1.52i)4-s + (−0.508 + 0.606i)5-s + (2.35 + 0.678i)6-s + (−3.72 − 1.35i)7-s + (−0.756 + 2.72i)8-s + (0.427 + 2.96i)9-s + (0.292 − 1.08i)10-s + (2.92 + 3.49i)11-s + (−3.42 + 0.527i)12-s + (3.31 + 0.584i)13-s + (5.57 − 0.471i)14-s + (1.35 − 0.216i)15-s + (−0.646 − 3.94i)16-s + (−2.50 + 4.34i)17-s + ⋯ |
| L(s) = 1 | + (−0.907 + 0.419i)2-s + (−0.755 − 0.654i)3-s + (0.647 − 0.762i)4-s + (−0.227 + 0.271i)5-s + (0.960 + 0.276i)6-s + (−1.40 − 0.511i)7-s + (−0.267 + 0.963i)8-s + (0.142 + 0.989i)9-s + (0.0926 − 0.341i)10-s + (0.883 + 1.05i)11-s + (−0.988 + 0.152i)12-s + (0.920 + 0.162i)13-s + (1.49 − 0.125i)14-s + (0.349 − 0.0559i)15-s + (−0.161 − 0.986i)16-s + (−0.608 + 1.05i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.424 - 0.905i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.424 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.164922 + 0.259409i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.164922 + 0.259409i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.28 - 0.593i)T \) |
| 3 | \( 1 + (1.30 + 1.13i)T \) |
| good | 5 | \( 1 + (0.508 - 0.606i)T + (-0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (3.72 + 1.35i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (-2.92 - 3.49i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-3.31 - 0.584i)T + (12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (2.50 - 4.34i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (6.10 - 3.52i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.33 - 1.57i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (3.32 - 0.585i)T + (27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-7.51 + 2.73i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (3.56 + 2.05i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.333 + 1.88i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-3.66 - 4.36i)T + (-7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (7.37 + 2.68i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + 1.95iT - 53T^{2} \) |
| 59 | \( 1 + (-3.02 + 3.60i)T + (-10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (1.87 - 5.15i)T + (-46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (5.36 + 0.945i)T + (62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (5.56 - 9.63i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (1.32 + 2.28i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.150 + 0.852i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (0.0489 - 0.00863i)T + (77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (2.15 + 3.73i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.00 + 4.19i)T + (16.8 - 95.5i)T^{2} \) |
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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
−12.55102390278593257903912840214, −11.50599627235616213066420832189, −10.53877722247724207142942718906, −9.849109582339243113677765032773, −8.590721931521765710344000953330, −7.40516659518976635720565789649, −6.47987132600267058960738376182, −6.12735947402863043373045248667, −4.03023103069522548255905444387, −1.75069172747370558699203272290,
0.37273006083646312686736223407, 3.03617940352311346503297527796, 4.19985375454598529919905694516, 6.21856842863930132100186484650, 6.56898531266480909226015809134, 8.592121310240677513734549668346, 9.069616906093272803120068886981, 10.07109300456640339209720938432, 10.99359514922843775751854986876, 11.78730776054034589682252774480
