L(s) = 1 | + (0.965 − 0.258i)3-s + (−2.25 − 1.38i)7-s + (0.866 − 0.499i)9-s + (−2.42 + 4.19i)11-s + (−2.27 − 2.27i)13-s + (−0.574 − 2.14i)17-s + (−0.107 − 0.185i)19-s + (−2.53 − 0.758i)21-s + (6.64 + 1.78i)23-s + (0.707 − 0.707i)27-s + 9.28i·29-s + (−7.78 − 4.49i)31-s + (−1.25 + 4.68i)33-s + (−2.62 + 9.78i)37-s + (−2.78 − 1.61i)39-s + ⋯ |
L(s) = 1 | + (0.557 − 0.149i)3-s + (−0.851 − 0.524i)7-s + (0.288 − 0.166i)9-s + (−0.730 + 1.26i)11-s + (−0.631 − 0.631i)13-s + (−0.139 − 0.520i)17-s + (−0.0246 − 0.0426i)19-s + (−0.553 − 0.165i)21-s + (1.38 + 0.371i)23-s + (0.136 − 0.136i)27-s + 1.72i·29-s + (−1.39 − 0.807i)31-s + (−0.218 + 0.815i)33-s + (−0.431 + 1.60i)37-s + (−0.446 − 0.257i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.753 - 0.657i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.753 - 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4518950337\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4518950337\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.965 + 0.258i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.25 + 1.38i)T \) |
good | 11 | \( 1 + (2.42 - 4.19i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.27 + 2.27i)T + 13iT^{2} \) |
| 17 | \( 1 + (0.574 + 2.14i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (0.107 + 0.185i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.64 - 1.78i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 - 9.28iT - 29T^{2} \) |
| 31 | \( 1 + (7.78 + 4.49i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.62 - 9.78i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + 2.32iT - 41T^{2} \) |
| 43 | \( 1 + (5.01 - 5.01i)T - 43iT^{2} \) |
| 47 | \( 1 + (9.36 + 2.50i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (0.785 + 2.93i)T + (-45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (2.68 - 4.64i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (13.2 - 7.63i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.97 + 1.33i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 + (-0.146 + 0.0391i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (0.0599 - 0.0346i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.467 - 0.467i)T + 83iT^{2} \) |
| 89 | \( 1 + (0.997 + 1.72i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.51 + 7.51i)T - 97iT^{2} \) |
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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
−9.433312203134912676154635389947, −8.736657195929885470172389660657, −7.52203141916160648659562575908, −7.32384694305189997292116461609, −6.52452008146352605874362859573, −5.21666505411971761978600329772, −4.66351086088492802555365175894, −3.35600422426523084780572560409, −2.82455682208335374705452783633, −1.54937179421926105736364480762,
0.13727979834743880588847154716, 1.99780586219474428086196733018, 2.96173197082517929432327530729, 3.60322152041875601791900286630, 4.77321962745982578529788275324, 5.66230865605605050103873846462, 6.43173245512855634831854125386, 7.30098049253141812843101386435, 8.152731230613810407499656342388, 8.943882854988360230983269485426