| 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
−14.63953529603477917678048342892, −12.90870571499192574564647767406, −11.77991568951202684283631669512, −10.84765973604870493145979788116, −9.919755662422736648627836904535, −8.808598535349014746064575290750, −8.084256617530878294407519711004, −5.77854475138926490770442247062, −4.86649898160977832508574133740, −1.55717632331102839278451878155,
1.55715856088049282814925727753, 4.91140310499595560865078058590, 6.81646742689473388577838871609, 7.27206567912929304012042339825, 8.617461726117951629419705285266, 10.19977589632730332958163802736, 10.80958018480574703062611971337, 11.73795710733769206406991924149, 13.65555029326117490155234056218, 14.15041661746865273726975915527
