L(s) = 1 | + (−0.336 − 2.33i)2-s + (1.10 − 2.42i)3-s + (−3.44 + 1.01i)4-s + (−6.04 − 1.77i)6-s + (2.96 − 1.90i)7-s + (1.55 + 3.41i)8-s + (−2.68 − 3.09i)9-s + (−0.318 + 2.21i)11-s + (−1.36 + 9.46i)12-s + (−2.99 − 1.92i)13-s + (−5.46 − 6.30i)14-s + (1.42 − 0.918i)16-s + (−5.12 − 1.50i)17-s + (−6.33 + 7.31i)18-s + (2.61 − 0.769i)19-s + ⋯ |
L(s) = 1 | + (−0.237 − 1.65i)2-s + (0.638 − 1.39i)3-s + (−1.72 + 0.505i)4-s + (−2.46 − 0.724i)6-s + (1.12 − 0.720i)7-s + (0.551 + 1.20i)8-s + (−0.893 − 1.03i)9-s + (−0.0959 + 0.667i)11-s + (−0.392 + 2.73i)12-s + (−0.831 − 0.534i)13-s + (−1.45 − 1.68i)14-s + (0.357 − 0.229i)16-s + (−1.24 − 0.365i)17-s + (−1.49 + 1.72i)18-s + (0.600 − 0.176i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.509 - 0.860i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.509 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.705929 + 1.23798i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.705929 + 1.23798i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 23 | \( 1 + (-1.35 - 4.60i)T \) |
good | 2 | \( 1 + (0.336 + 2.33i)T + (-1.91 + 0.563i)T^{2} \) |
| 3 | \( 1 + (-1.10 + 2.42i)T + (-1.96 - 2.26i)T^{2} \) |
| 7 | \( 1 + (-2.96 + 1.90i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (0.318 - 2.21i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (2.99 + 1.92i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (5.12 + 1.50i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (-2.61 + 0.769i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (-4.25 - 1.24i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (0.941 + 2.06i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (4.24 + 4.90i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (-1.88 + 2.17i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (-1.67 + 3.66i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 + (-6.11 + 3.93i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (-5.02 - 3.23i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (1.60 + 3.51i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (-0.217 - 1.51i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (0.0271 + 0.188i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (4.32 - 1.27i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (4.72 + 3.03i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (-4.89 - 5.64i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (2.09 - 4.57i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (6.12 - 7.07i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.36200225116828441752330732416, −9.310575934314264861718674404403, −8.524622310564285895219188577323, −7.51154433892875156093543072372, −7.07184539053120632269225570988, −5.12726717235409873396179478589, −4.04390780597287632301825750421, −2.64824714742357195903013269595, −1.94943137639509041542173193550, −0.844233629328679510170402021081,
2.61063158080966429485703081425, 4.32662969530308997847334965587, 4.84477595843234812349365309800, 5.72556150827546794029823967012, 6.88337326129319088798107098856, 8.017786864073962862577908597377, 8.732065552352028195010866555619, 9.017052983407830183666233604373, 10.09372661088366764687487594962, 11.02097322730267229939145928465