| L(s) = 1 | + (−1.86 + 0.441i)2-s + (−1.43 − 0.975i)3-s + (1.48 − 0.744i)4-s + (1.48 + 1.99i)5-s + (3.09 + 1.18i)6-s + (3.80 − 2.50i)7-s + (0.501 − 0.420i)8-s + (1.09 + 2.79i)9-s + (−3.64 − 3.05i)10-s + (−0.315 − 0.0368i)11-s + (−2.84 − 0.380i)12-s + (0.975 + 3.25i)13-s + (−5.97 + 6.33i)14-s + (−0.179 − 4.29i)15-s + (−2.72 + 3.66i)16-s + (0.885 − 5.02i)17-s + ⋯ |
| L(s) = 1 | + (−1.31 + 0.311i)2-s + (−0.826 − 0.563i)3-s + (0.740 − 0.372i)4-s + (0.663 + 0.891i)5-s + (1.26 + 0.483i)6-s + (1.43 − 0.945i)7-s + (0.177 − 0.148i)8-s + (0.365 + 0.930i)9-s + (−1.15 − 0.965i)10-s + (−0.0951 − 0.0111i)11-s + (−0.821 − 0.109i)12-s + (0.270 + 0.903i)13-s + (−1.59 + 1.69i)14-s + (−0.0462 − 1.11i)15-s + (−0.681 + 0.915i)16-s + (0.214 − 1.21i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.127i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.504235 + 0.0322226i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.504235 + 0.0322226i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.43 + 0.975i)T \) |
| good | 2 | \( 1 + (1.86 - 0.441i)T + (1.78 - 0.897i)T^{2} \) |
| 5 | \( 1 + (-1.48 - 1.99i)T + (-1.43 + 4.78i)T^{2} \) |
| 7 | \( 1 + (-3.80 + 2.50i)T + (2.77 - 6.42i)T^{2} \) |
| 11 | \( 1 + (0.315 + 0.0368i)T + (10.7 + 2.53i)T^{2} \) |
| 13 | \( 1 + (-0.975 - 3.25i)T + (-10.8 + 7.14i)T^{2} \) |
| 17 | \( 1 + (-0.885 + 5.02i)T + (-15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (0.216 + 1.22i)T + (-17.8 + 6.49i)T^{2} \) |
| 23 | \( 1 + (-1.59 - 1.04i)T + (9.10 + 21.1i)T^{2} \) |
| 29 | \( 1 + (-2.37 - 2.52i)T + (-1.68 + 28.9i)T^{2} \) |
| 31 | \( 1 + (0.394 - 6.77i)T + (-30.7 - 3.59i)T^{2} \) |
| 37 | \( 1 + (8.02 - 2.91i)T + (28.3 - 23.7i)T^{2} \) |
| 41 | \( 1 + (6.17 + 1.46i)T + (36.6 + 18.4i)T^{2} \) |
| 43 | \( 1 + (2.41 + 5.59i)T + (-29.5 + 31.2i)T^{2} \) |
| 47 | \( 1 + (0.237 + 4.07i)T + (-46.6 + 5.45i)T^{2} \) |
| 53 | \( 1 + (3.88 - 6.72i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.94 - 0.460i)T + (57.4 - 13.6i)T^{2} \) |
| 61 | \( 1 + (-2.19 - 1.10i)T + (36.4 + 48.9i)T^{2} \) |
| 67 | \( 1 + (-2.36 + 2.50i)T + (-3.89 - 66.8i)T^{2} \) |
| 71 | \( 1 + (2.16 + 1.81i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (3.24 - 2.71i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (7.04 - 1.67i)T + (70.5 - 35.4i)T^{2} \) |
| 83 | \( 1 + (11.7 - 2.78i)T + (74.1 - 37.2i)T^{2} \) |
| 89 | \( 1 + (-2.86 + 2.40i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-8.73 + 11.7i)T + (-27.8 - 92.9i)T^{2} \) |
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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
−14.15041661746865273726975915527, −13.65555029326117490155234056218, −11.73795710733769206406991924149, −10.80958018480574703062611971337, −10.19977589632730332958163802736, −8.617461726117951629419705285266, −7.27206567912929304012042339825, −6.81646742689473388577838871609, −4.91140310499595560865078058590, −1.55715856088049282814925727753,
1.55717632331102839278451878155, 4.86649898160977832508574133740, 5.77854475138926490770442247062, 8.084256617530878294407519711004, 8.808598535349014746064575290750, 9.919755662422736648627836904535, 10.84765973604870493145979788116, 11.77991568951202684283631669512, 12.90870571499192574564647767406, 14.63953529603477917678048342892
