| L(s) = 1 | + (−0.643 − 0.152i)2-s + (−1.70 − 0.278i)3-s + (−1.39 − 0.701i)4-s + (−1.50 + 2.01i)5-s + (1.05 + 0.440i)6-s + (−3.43 − 2.25i)7-s + (1.80 + 1.51i)8-s + (2.84 + 0.953i)9-s + (1.27 − 1.06i)10-s + (−2.94 + 0.343i)11-s + (2.19 + 1.58i)12-s + (0.930 − 3.10i)13-s + (1.86 + 1.97i)14-s + (3.12 − 3.02i)15-s + (0.935 + 1.25i)16-s + (−0.572 − 3.24i)17-s + ⋯ |
| L(s) = 1 | + (−0.455 − 0.107i)2-s + (−0.986 − 0.160i)3-s + (−0.698 − 0.350i)4-s + (−0.671 + 0.901i)5-s + (0.431 + 0.179i)6-s + (−1.29 − 0.854i)7-s + (0.638 + 0.535i)8-s + (0.948 + 0.317i)9-s + (0.402 − 0.337i)10-s + (−0.886 + 0.103i)11-s + (0.632 + 0.458i)12-s + (0.257 − 0.861i)13-s + (0.498 + 0.528i)14-s + (0.807 − 0.781i)15-s + (0.233 + 0.314i)16-s + (−0.138 − 0.787i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0260i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0260i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.000655225 + 0.0502352i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.000655225 + 0.0502352i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.70 + 0.278i)T \) |
| good | 2 | \( 1 + (0.643 + 0.152i)T + (1.78 + 0.897i)T^{2} \) |
| 5 | \( 1 + (1.50 - 2.01i)T + (-1.43 - 4.78i)T^{2} \) |
| 7 | \( 1 + (3.43 + 2.25i)T + (2.77 + 6.42i)T^{2} \) |
| 11 | \( 1 + (2.94 - 0.343i)T + (10.7 - 2.53i)T^{2} \) |
| 13 | \( 1 + (-0.930 + 3.10i)T + (-10.8 - 7.14i)T^{2} \) |
| 17 | \( 1 + (0.572 + 3.24i)T + (-15.9 + 5.81i)T^{2} \) |
| 19 | \( 1 + (0.571 - 3.23i)T + (-17.8 - 6.49i)T^{2} \) |
| 23 | \( 1 + (2.08 - 1.37i)T + (9.10 - 21.1i)T^{2} \) |
| 29 | \( 1 + (3.82 - 4.04i)T + (-1.68 - 28.9i)T^{2} \) |
| 31 | \( 1 + (0.373 + 6.41i)T + (-30.7 + 3.59i)T^{2} \) |
| 37 | \( 1 + (2.56 + 0.935i)T + (28.3 + 23.7i)T^{2} \) |
| 41 | \( 1 + (8.80 - 2.08i)T + (36.6 - 18.4i)T^{2} \) |
| 43 | \( 1 + (3.09 - 7.17i)T + (-29.5 - 31.2i)T^{2} \) |
| 47 | \( 1 + (-0.588 + 10.1i)T + (-46.6 - 5.45i)T^{2} \) |
| 53 | \( 1 + (-0.00494 - 0.00856i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-11.2 - 1.32i)T + (57.4 + 13.6i)T^{2} \) |
| 61 | \( 1 + (-8.05 + 4.04i)T + (36.4 - 48.9i)T^{2} \) |
| 67 | \( 1 + (4.01 + 4.25i)T + (-3.89 + 66.8i)T^{2} \) |
| 71 | \( 1 + (1.81 - 1.52i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-3.61 - 3.03i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-8.75 - 2.07i)T + (70.5 + 35.4i)T^{2} \) |
| 83 | \( 1 + (3.60 + 0.853i)T + (74.1 + 37.2i)T^{2} \) |
| 89 | \( 1 + (9.57 + 8.03i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (4.27 + 5.73i)T + (-27.8 + 92.9i)T^{2} \) |
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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
−13.52815850731764190747091550806, −12.87966138196750626895165328834, −11.34113846395672304173882300096, −10.36986105057791226903412966275, −9.882730207448731895856204562647, −7.87068472246151592588431585301, −6.87107736163714762411628153905, −5.43070228887298366550604833933, −3.69446049519770688848837369930, −0.082403625574968945261391313953,
3.97426229494947667271562729036, 5.27028505967629612572073876517, 6.75571361635562234741996510890, 8.377146701752960322685266735302, 9.254281738065730297918046788905, 10.36053171714956120412453220798, 11.90846532976535790111658256751, 12.65509943223143568119672102303, 13.34281958122028481448177815074, 15.50273361967124579840424818591
