| L(s) = 1 | + (0.210 − 1.39i)2-s + (1.12 + 1.31i)3-s + (−1.91 − 0.589i)4-s + (−0.614 + 0.700i)5-s + (2.07 − 1.29i)6-s + (−2.61 + 0.344i)7-s + (−1.22 + 2.54i)8-s + (−0.469 + 2.96i)9-s + (0.850 + 1.00i)10-s + (−0.249 + 0.506i)11-s + (−1.37 − 3.17i)12-s + (−0.126 + 0.371i)13-s + (−0.0693 + 3.72i)14-s + (−1.61 − 0.0211i)15-s + (3.30 + 2.25i)16-s + (−2.14 + 2.14i)17-s + ⋯ |
| L(s) = 1 | + (0.148 − 0.988i)2-s + (0.649 + 0.760i)3-s + (−0.955 − 0.294i)4-s + (−0.274 + 0.313i)5-s + (0.848 − 0.528i)6-s + (−0.988 + 0.130i)7-s + (−0.433 + 0.901i)8-s + (−0.156 + 0.987i)9-s + (0.268 + 0.318i)10-s + (−0.0753 + 0.152i)11-s + (−0.396 − 0.917i)12-s + (−0.0349 + 0.103i)13-s + (−0.0185 + 0.996i)14-s + (−0.416 − 0.00544i)15-s + (0.826 + 0.562i)16-s + (−0.520 + 0.520i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.254 - 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.254 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.816923 + 0.629474i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.816923 + 0.629474i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.210 + 1.39i)T \) |
| 3 | \( 1 + (-1.12 - 1.31i)T \) |
| good | 5 | \( 1 + (0.614 - 0.700i)T + (-0.652 - 4.95i)T^{2} \) |
| 7 | \( 1 + (2.61 - 0.344i)T + (6.76 - 1.81i)T^{2} \) |
| 11 | \( 1 + (0.249 - 0.506i)T + (-6.69 - 8.72i)T^{2} \) |
| 13 | \( 1 + (0.126 - 0.371i)T + (-10.3 - 7.91i)T^{2} \) |
| 17 | \( 1 + (2.14 - 2.14i)T - 17iT^{2} \) |
| 19 | \( 1 + (-1.03 - 5.18i)T + (-17.5 + 7.27i)T^{2} \) |
| 23 | \( 1 + (-0.0230 + 0.175i)T + (-22.2 - 5.95i)T^{2} \) |
| 29 | \( 1 + (-3.44 - 0.226i)T + (28.7 + 3.78i)T^{2} \) |
| 31 | \( 1 + (6.55 + 3.78i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.453 - 2.28i)T + (-34.1 - 14.1i)T^{2} \) |
| 41 | \( 1 + (0.149 - 1.13i)T + (-39.6 - 10.6i)T^{2} \) |
| 43 | \( 1 + (4.73 + 2.33i)T + (26.1 + 34.1i)T^{2} \) |
| 47 | \( 1 + (-5.24 - 1.40i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-1.39 - 2.08i)T + (-20.2 + 48.9i)T^{2} \) |
| 59 | \( 1 + (-5.10 + 5.81i)T + (-7.70 - 58.4i)T^{2} \) |
| 61 | \( 1 + (-0.478 + 7.30i)T + (-60.4 - 7.96i)T^{2} \) |
| 67 | \( 1 + (5.64 - 2.78i)T + (40.7 - 53.1i)T^{2} \) |
| 71 | \( 1 + (-2.68 + 6.48i)T + (-50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (-3.49 - 8.43i)T + (-51.6 + 51.6i)T^{2} \) |
| 79 | \( 1 + (-0.114 + 0.426i)T + (-68.4 - 39.5i)T^{2} \) |
| 83 | \( 1 + (-8.40 + 7.37i)T + (10.8 - 82.2i)T^{2} \) |
| 89 | \( 1 + (-13.4 - 5.58i)T + (62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 + (-10.4 + 6.01i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.74491170575236466617522171380, −10.06785094172973123992017005594, −9.400993288400867076292000043539, −8.618174346904304811375881350015, −7.60829551560608536216108307220, −6.15548876872129762447528222345, −5.02387497245426899036560240194, −3.83023348154418540105344306404, −3.29912424516154572318855590373, −2.07347570980056447386554109047,
0.50496018731665444913768570099, 2.78940738092312323701462143655, 3.83623391799015842039217746435, 5.07012370092301020235045301325, 6.32559173104313325939789513837, 6.94631268283440723837460258211, 7.70437750825093518457573925875, 8.793602525783019136027785068047, 9.143528969925663211041088534237, 10.24594295048163266802226533882
