| L(s) = 1 | + (−0.930 + 1.96i)2-s + (0.189 + 0.174i)3-s + (−1.71 − 2.10i)4-s + (−2.11 − 0.730i)5-s + (−0.517 + 0.208i)6-s + (1.06 + 0.308i)7-s + (1.49 − 0.369i)8-s + (−0.236 − 2.92i)9-s + (3.39 − 3.46i)10-s + (2.20 + 2.59i)11-s + (0.0421 − 0.696i)12-s + (−3.58 − 0.385i)13-s + (−1.59 + 1.79i)14-s + (−0.272 − 0.506i)15-s + (0.414 − 2.02i)16-s + (0.0735 + 0.133i)17-s + ⋯ |
| L(s) = 1 | + (−0.657 + 1.38i)2-s + (0.109 + 0.100i)3-s + (−0.857 − 1.05i)4-s + (−0.945 − 0.326i)5-s + (−0.211 + 0.0851i)6-s + (0.402 + 0.116i)7-s + (0.530 − 0.130i)8-s + (−0.0786 − 0.974i)9-s + (1.07 − 1.09i)10-s + (0.665 + 0.782i)11-s + (0.0121 − 0.200i)12-s + (−0.994 − 0.106i)13-s + (−0.425 + 0.480i)14-s + (−0.0702 − 0.130i)15-s + (0.103 − 0.507i)16-s + (0.0178 + 0.0324i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0645 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0645 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.661907 + 0.620456i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.661907 + 0.620456i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (2.11 + 0.730i)T \) |
| 13 | \( 1 + (3.58 + 0.385i)T \) |
| good | 2 | \( 1 + (0.930 - 1.96i)T + (-1.26 - 1.54i)T^{2} \) |
| 3 | \( 1 + (-0.189 - 0.174i)T + (0.241 + 2.99i)T^{2} \) |
| 7 | \( 1 + (-1.06 - 0.308i)T + (5.91 + 3.74i)T^{2} \) |
| 11 | \( 1 + (-2.20 - 2.59i)T + (-1.76 + 10.8i)T^{2} \) |
| 17 | \( 1 + (-0.0735 - 0.133i)T + (-9.08 + 14.3i)T^{2} \) |
| 19 | \( 1 + (-5.69 + 1.52i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-0.410 + 1.53i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (-3.99 - 1.89i)T + (18.3 + 22.4i)T^{2} \) |
| 31 | \( 1 + (-3.70 - 2.89i)T + (7.41 + 30.0i)T^{2} \) |
| 37 | \( 1 + (4.52 + 6.01i)T + (-10.2 + 35.5i)T^{2} \) |
| 41 | \( 1 + (-8.64 - 7.97i)T + (3.29 + 40.8i)T^{2} \) |
| 43 | \( 1 + (-0.768 - 5.41i)T + (-41.3 + 11.9i)T^{2} \) |
| 47 | \( 1 + (-3.90 + 1.47i)T + (35.1 - 31.1i)T^{2} \) |
| 53 | \( 1 + (4.69 - 7.76i)T + (-24.6 - 46.9i)T^{2} \) |
| 59 | \( 1 + (-9.45 - 6.25i)T + (23.1 + 54.2i)T^{2} \) |
| 61 | \( 1 + (0.00257 - 0.00267i)T + (-2.45 - 60.9i)T^{2} \) |
| 67 | \( 1 + (-6.39 + 7.83i)T + (-13.4 - 65.6i)T^{2} \) |
| 71 | \( 1 + (-2.08 + 9.24i)T + (-64.1 - 30.4i)T^{2} \) |
| 73 | \( 1 + (0.0844 - 0.122i)T + (-25.8 - 68.2i)T^{2} \) |
| 79 | \( 1 + (-13.4 + 5.11i)T + (59.1 - 52.3i)T^{2} \) |
| 83 | \( 1 + (-4.24 - 8.08i)T + (-47.1 + 68.3i)T^{2} \) |
| 89 | \( 1 + (14.0 + 3.77i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-10.2 + 3.41i)T + (77.5 - 58.2i)T^{2} \) |
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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
−9.860246041727611761603995654766, −9.308170871723351056300060114718, −8.607088183434719653519146257678, −7.71671124268123973880041517021, −7.16388153830060289804990858728, −6.38578701927250211056765222554, −5.15353768439607874583433261036, −4.43728290895987732780688161571, −3.06918148842481143637418451176, −0.855682461941121019393189127006,
0.863356317772413943777826558347, 2.28657038589614828769955642994, 3.23029539409886434537330294506, 4.16946146758556830532564547168, 5.32215769606941740799340731156, 6.83784685515240780456470599943, 7.85223244992516870456015679104, 8.320510937080510423725354320395, 9.343876213625590460091725274265, 10.16091575457773411090325075576
