| 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
−10.82798454780890473015976514338, −9.555385831813765870610410167808, −9.334867769668637733146768306295, −8.273993261539817358388968528125, −7.23620093719234419498316179879, −6.48658072334559268963937296522, −5.24554756607688509024934439609, −4.32752034982353615600276648206, −3.34091248877854548664833828161, −2.60960764503257882055475974496,
0.14451853512218194869432485226, 1.58866017871108620245314036453, 3.14333470434256971662746009527, 3.94268840526430403103004581184, 5.39771566405284702589175231941, 6.17950913147061475738652571513, 7.27170624946579826425617639604, 7.963070030611138455006683005254, 8.484529526787353221653802856468, 9.560873774776454283941989837291
