| L(s) = 1 | + (0.884 − 0.547i)2-s + (0.273 + 0.961i)3-s + (−0.409 + 0.821i)4-s + (3.15 − 1.22i)5-s + (0.768 + 0.700i)6-s + (0.393 − 4.25i)7-s + (0.280 + 3.02i)8-s + (−0.850 + 0.526i)9-s + (2.11 − 2.80i)10-s + (−0.740 + 0.458i)11-s + (−0.901 − 0.168i)12-s + (−0.339 + 3.65i)13-s + (−1.98 − 3.97i)14-s + (2.03 + 2.69i)15-s + (0.797 + 1.05i)16-s + (0.989 − 0.902i)17-s + ⋯ |
| L(s) = 1 | + (0.625 − 0.387i)2-s + (0.157 + 0.555i)3-s + (−0.204 + 0.410i)4-s + (1.40 − 0.546i)5-s + (0.313 + 0.286i)6-s + (0.148 − 1.60i)7-s + (0.0990 + 1.06i)8-s + (−0.283 + 0.175i)9-s + (0.670 − 0.887i)10-s + (−0.223 + 0.138i)11-s + (−0.260 − 0.0486i)12-s + (−0.0940 + 1.01i)13-s + (−0.529 − 1.06i)14-s + (0.526 + 0.696i)15-s + (0.199 + 0.263i)16-s + (0.240 − 0.218i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0730i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0730i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.09845 - 0.0767946i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.09845 - 0.0767946i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.273 - 0.961i)T \) |
| 103 | \( 1 + (-5.32 + 8.64i)T \) |
| good | 2 | \( 1 + (-0.884 + 0.547i)T + (0.891 - 1.79i)T^{2} \) |
| 5 | \( 1 + (-3.15 + 1.22i)T + (3.69 - 3.36i)T^{2} \) |
| 7 | \( 1 + (-0.393 + 4.25i)T + (-6.88 - 1.28i)T^{2} \) |
| 11 | \( 1 + (0.740 - 0.458i)T + (4.90 - 9.84i)T^{2} \) |
| 13 | \( 1 + (0.339 - 3.65i)T + (-12.7 - 2.38i)T^{2} \) |
| 17 | \( 1 + (-0.989 + 0.902i)T + (1.56 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-0.321 + 1.12i)T + (-16.1 - 10.0i)T^{2} \) |
| 23 | \( 1 + (0.142 + 0.0882i)T + (10.2 + 20.5i)T^{2} \) |
| 29 | \( 1 + (5.84 - 2.26i)T + (21.4 - 19.5i)T^{2} \) |
| 31 | \( 1 + (-1.70 - 2.25i)T + (-8.48 + 29.8i)T^{2} \) |
| 37 | \( 1 + (-0.129 - 0.0242i)T + (34.5 + 13.3i)T^{2} \) |
| 41 | \( 1 + (-0.758 - 0.293i)T + (30.2 + 27.6i)T^{2} \) |
| 43 | \( 1 + (4.12 - 0.770i)T + (40.0 - 15.5i)T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 + (-3.43 + 12.0i)T + (-45.0 - 27.9i)T^{2} \) |
| 59 | \( 1 + (1.00 + 10.8i)T + (-57.9 + 10.8i)T^{2} \) |
| 61 | \( 1 + (10.7 - 9.76i)T + (5.62 - 60.7i)T^{2} \) |
| 67 | \( 1 + (-0.168 - 1.81i)T + (-65.8 + 12.3i)T^{2} \) |
| 71 | \( 1 + (9.84 + 3.81i)T + (52.4 + 47.8i)T^{2} \) |
| 73 | \( 1 + (-12.7 - 4.92i)T + (53.9 + 49.1i)T^{2} \) |
| 79 | \( 1 + (-9.76 + 3.78i)T + (58.3 - 53.2i)T^{2} \) |
| 83 | \( 1 + (0.475 - 5.13i)T + (-81.5 - 15.2i)T^{2} \) |
| 89 | \( 1 + (-4.05 - 8.13i)T + (-53.6 + 71.0i)T^{2} \) |
| 97 | \( 1 + (-13.2 - 12.0i)T + (8.95 + 96.5i)T^{2} \) |
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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
−11.66790254004477451761004543143, −10.74322562834518928414238336515, −9.842287624774481513766333313360, −9.112676060092459037963937803051, −7.941142624544447789699623645392, −6.72332377306514911783618252197, −5.22250625212091203695184612599, −4.54999517000971777717061055033, −3.46163834502257192369290040821, −1.86100731244721597540067417643,
1.87958054452219512900156944146, 3.04686951590400706028873392595, 5.15126214374920070423800027874, 5.87161420803213663207315793032, 6.27908891311941217290994368410, 7.74777821296522714431688364271, 9.004177895659318733950502759047, 9.753784533759162893570948667388, 10.65161889343361143529520257148, 12.00041483797921786117611032851
