| L(s) = 1 | + (0.0177 + 0.112i)2-s + (1.39 − 1.02i)3-s + (1.88 − 0.614i)4-s + (−0.831 − 0.423i)5-s + (0.139 + 0.138i)6-s + (−3.16 + 0.759i)7-s + (0.205 + 0.403i)8-s + (0.910 − 2.85i)9-s + (0.0327 − 0.100i)10-s + (−3.17 + 3.72i)11-s + (2.01 − 2.79i)12-s + (1.36 + 2.23i)13-s + (−0.141 − 0.341i)14-s + (−1.59 + 0.257i)15-s + (3.17 − 2.30i)16-s + (2.48 − 0.195i)17-s + ⋯ |
| L(s) = 1 | + (0.0125 + 0.0793i)2-s + (0.807 − 0.590i)3-s + (0.944 − 0.307i)4-s + (−0.371 − 0.189i)5-s + (0.0569 + 0.0566i)6-s + (−1.19 + 0.287i)7-s + (0.0726 + 0.142i)8-s + (0.303 − 0.952i)9-s + (0.0103 − 0.0318i)10-s + (−0.957 + 1.12i)11-s + (0.581 − 0.805i)12-s + (0.379 + 0.619i)13-s + (−0.0377 − 0.0912i)14-s + (−0.412 + 0.0665i)15-s + (0.793 − 0.576i)16-s + (0.602 − 0.0474i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.885 + 0.465i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.885 + 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.32552 - 0.327208i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.32552 - 0.327208i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.39 + 1.02i)T \) |
| 41 | \( 1 + (-0.359 + 6.39i)T \) |
| good | 2 | \( 1 + (-0.0177 - 0.112i)T + (-1.90 + 0.618i)T^{2} \) |
| 5 | \( 1 + (0.831 + 0.423i)T + (2.93 + 4.04i)T^{2} \) |
| 7 | \( 1 + (3.16 - 0.759i)T + (6.23 - 3.17i)T^{2} \) |
| 11 | \( 1 + (3.17 - 3.72i)T + (-1.72 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.36 - 2.23i)T + (-5.90 + 11.5i)T^{2} \) |
| 17 | \( 1 + (-2.48 + 0.195i)T + (16.7 - 2.65i)T^{2} \) |
| 19 | \( 1 + (0.671 - 1.09i)T + (-8.62 - 16.9i)T^{2} \) |
| 23 | \( 1 + (-6.78 - 4.92i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (8.97 + 0.706i)T + (28.6 + 4.53i)T^{2} \) |
| 31 | \( 1 + (6.12 + 1.99i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-0.315 - 0.970i)T + (-29.9 + 21.7i)T^{2} \) |
| 43 | \( 1 + (-4.84 + 0.767i)T + (40.8 - 13.2i)T^{2} \) |
| 47 | \( 1 + (-0.231 + 0.965i)T + (-41.8 - 21.3i)T^{2} \) |
| 53 | \( 1 + (-0.432 + 5.49i)T + (-52.3 - 8.29i)T^{2} \) |
| 59 | \( 1 + (-1.20 + 1.66i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (0.214 - 1.35i)T + (-58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (-2.71 - 3.18i)T + (-10.4 + 66.1i)T^{2} \) |
| 71 | \( 1 + (-1.10 - 0.942i)T + (11.1 + 70.1i)T^{2} \) |
| 73 | \( 1 + (-0.923 + 0.923i)T - 73iT^{2} \) |
| 79 | \( 1 + (5.16 - 12.4i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + 7.89iT - 83T^{2} \) |
| 89 | \( 1 + (2.73 + 11.4i)T + (-79.2 + 40.4i)T^{2} \) |
| 97 | \( 1 + (-11.4 + 9.80i)T + (15.1 - 95.8i)T^{2} \) |
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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
−13.10108924315980516170815052265, −12.56473981112228474711331532421, −11.46644051299284013916880385329, −10.03825113833775894148220392397, −9.173155291822903637133421935502, −7.62833503392018899049119011869, −7.02714741341835399614376325905, −5.72249749723150915962994149406, −3.52268027272138085149758277518, −2.12667148557902195594313624905,
2.92760581186436439602589893099, 3.54176996431228448795681125924, 5.69019842178121154729719676681, 7.16945531014529219753622927685, 8.069302238744144541253801659765, 9.315348998053382874047714305116, 10.64327154256382245426791210907, 11.02556374420881828523023467823, 12.81068618819956388044101601167, 13.25907189950475592706946196332
