| L(s) = 1 | + (−1.35 + 0.395i)2-s + (0.178 − 0.489i)3-s + (1.68 − 1.07i)4-s + (−3.55 + 0.626i)5-s + (−0.0483 + 0.735i)6-s + (−1.81 + 3.13i)7-s + (−1.86 + 2.12i)8-s + (2.09 + 1.75i)9-s + (4.57 − 2.25i)10-s + (0.380 − 0.219i)11-s + (−0.225 − 1.01i)12-s + (1.07 + 2.95i)13-s + (1.21 − 4.97i)14-s + (−0.326 + 1.85i)15-s + (1.69 − 3.62i)16-s + (−5.14 + 4.31i)17-s + ⋯ |
| L(s) = 1 | + (−0.960 + 0.279i)2-s + (0.102 − 0.282i)3-s + (0.843 − 0.536i)4-s + (−1.58 + 0.280i)5-s + (−0.0197 + 0.300i)6-s + (−0.684 + 1.18i)7-s + (−0.659 + 0.751i)8-s + (0.696 + 0.584i)9-s + (1.44 − 0.713i)10-s + (0.114 − 0.0661i)11-s + (−0.0650 − 0.293i)12-s + (0.298 + 0.820i)13-s + (0.325 − 1.32i)14-s + (−0.0843 + 0.478i)15-s + (0.423 − 0.906i)16-s + (−1.24 + 1.04i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.549 - 0.835i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.549 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.191314 + 0.354974i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.191314 + 0.354974i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.35 - 0.395i)T \) |
| 19 | \( 1 + (4.34 + 0.341i)T \) |
| good | 3 | \( 1 + (-0.178 + 0.489i)T + (-2.29 - 1.92i)T^{2} \) |
| 5 | \( 1 + (3.55 - 0.626i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (1.81 - 3.13i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.380 + 0.219i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.07 - 2.95i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (5.14 - 4.31i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-0.958 + 5.43i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (1.67 - 1.99i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-2.59 + 4.49i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 5.95iT - 37T^{2} \) |
| 41 | \( 1 + (-0.621 - 0.226i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.376 + 0.0663i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (3.54 + 2.97i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-13.0 - 2.30i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-8.50 - 10.1i)T + (-10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (5.02 + 0.885i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (0.576 - 0.687i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.20 - 6.83i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-3.14 - 1.14i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (6.23 + 2.26i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (5.47 + 3.15i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.76 + 1.00i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (3.66 - 3.07i)T + (16.8 - 95.5i)T^{2} \) |
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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
−13.07976930445187485612071135647, −12.11021561097669062015196222577, −11.27090221789933484015272042486, −10.33539356789405556731780324920, −8.793022199399333448377665990771, −8.367206448513527197794972545001, −7.05909757200144097958696208083, −6.32112729068614150005573898239, −4.26670539424602107283907618547, −2.37611711396770318016921621571,
0.51833487277416283240189962022, 3.42948236845403525591873482305, 4.22590728797951938045127054788, 6.80561882655940336057117350357, 7.42025424214104302687818479116, 8.544683605826193993695748438036, 9.588834284857961802257650925448, 10.62362304506305006761365533872, 11.40341783626843979212889179348, 12.47790143003655937018258580024
