| L(s) = 1 | + (0.381 − 1.68i)3-s + (−1.75 − 1.47i)5-s + (−1.87 − 0.682i)7-s + (−2.70 − 1.28i)9-s + (−1.07 + 0.902i)11-s + (0.725 − 4.11i)13-s + (−3.15 + 2.40i)15-s + (−2.25 + 3.90i)17-s + (3.63 + 6.29i)19-s + (−1.86 + 2.90i)21-s + (−0.863 + 0.314i)23-s + (0.0416 + 0.236i)25-s + (−3.21 + 4.08i)27-s + (0.312 + 1.76i)29-s + (−7.27 + 2.64i)31-s + ⋯ |
| L(s) = 1 | + (0.220 − 0.975i)3-s + (−0.784 − 0.658i)5-s + (−0.708 − 0.258i)7-s + (−0.902 − 0.429i)9-s + (−0.324 + 0.271i)11-s + (0.201 − 1.14i)13-s + (−0.814 + 0.620i)15-s + (−0.546 + 0.945i)17-s + (0.833 + 1.44i)19-s + (−0.407 + 0.634i)21-s + (−0.180 + 0.0655i)23-s + (0.00833 + 0.0472i)25-s + (−0.618 + 0.786i)27-s + (0.0579 + 0.328i)29-s + (−1.30 + 0.475i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.709 - 0.705i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.709 - 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.119959 + 0.290776i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.119959 + 0.290776i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.381 + 1.68i)T \) |
| good | 5 | \( 1 + (1.75 + 1.47i)T + (0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (1.87 + 0.682i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (1.07 - 0.902i)T + (1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.725 + 4.11i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (2.25 - 3.90i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.63 - 6.29i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.863 - 0.314i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.312 - 1.76i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (7.27 - 2.64i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (-5.34 + 9.26i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.315 - 1.78i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (4.27 - 3.58i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (4.45 + 1.62i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + 2.86T + 53T^{2} \) |
| 59 | \( 1 + (0.648 + 0.544i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-2.05 - 0.748i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.71 + 9.75i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (7.17 - 12.4i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.64 - 2.85i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.28 + 12.9i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-0.874 - 4.95i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (9.29 + 16.1i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.10 + 5.12i)T + (16.8 - 95.5i)T^{2} \) |
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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
−9.560873774776454283941989837291, −8.484529526787353221653802856468, −7.963070030611138455006683005254, −7.27170624946579826425617639604, −6.17950913147061475738652571513, −5.39771566405284702589175231941, −3.94268840526430403103004581184, −3.14333470434256971662746009527, −1.58866017871108620245314036453, −0.14451853512218194869432485226,
2.60960764503257882055475974496, 3.34091248877854548664833828161, 4.32752034982353615600276648206, 5.24554756607688509024934439609, 6.48658072334559268963937296522, 7.23620093719234419498316179879, 8.273993261539817358388968528125, 9.334867769668637733146768306295, 9.555385831813765870610410167808, 10.82798454780890473015976514338
