L(s) = 1 | + (0.222 + 0.974i)2-s + (−0.273 + 1.19i)3-s + (−0.900 + 0.433i)4-s + (−0.623 − 0.781i)5-s − 1.22·6-s − 4.39·7-s + (−0.623 − 0.781i)8-s + (1.34 + 0.647i)9-s + (0.623 − 0.781i)10-s + (−2.14 − 1.03i)11-s + (−0.273 − 1.19i)12-s + (−2.19 − 2.75i)13-s + (−0.978 − 4.28i)14-s + (1.10 − 0.532i)15-s + (0.623 − 0.781i)16-s + (0.329 − 0.412i)17-s + ⋯ |
L(s) = 1 | + (0.157 + 0.689i)2-s + (−0.157 + 0.690i)3-s + (−0.450 + 0.216i)4-s + (−0.278 − 0.349i)5-s − 0.501·6-s − 1.66·7-s + (−0.220 − 0.276i)8-s + (0.448 + 0.215i)9-s + (0.197 − 0.247i)10-s + (−0.647 − 0.311i)11-s + (−0.0788 − 0.345i)12-s + (−0.609 − 0.764i)13-s + (−0.261 − 1.14i)14-s + (0.285 − 0.137i)15-s + (0.155 − 0.195i)16-s + (0.0798 − 0.100i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.430 + 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.430 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0464987 - 0.0737097i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0464987 - 0.0737097i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.222 - 0.974i)T \) |
| 5 | \( 1 + (0.623 + 0.781i)T \) |
| 43 | \( 1 + (-3.01 + 5.82i)T \) |
good | 3 | \( 1 + (0.273 - 1.19i)T + (-2.70 - 1.30i)T^{2} \) |
| 7 | \( 1 + 4.39T + 7T^{2} \) |
| 11 | \( 1 + (2.14 + 1.03i)T + (6.85 + 8.60i)T^{2} \) |
| 13 | \( 1 + (2.19 + 2.75i)T + (-2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (-0.329 + 0.412i)T + (-3.78 - 16.5i)T^{2} \) |
| 19 | \( 1 + (2.22 - 1.07i)T + (11.8 - 14.8i)T^{2} \) |
| 23 | \( 1 + (2.56 + 1.23i)T + (14.3 + 17.9i)T^{2} \) |
| 29 | \( 1 + (0.718 + 3.14i)T + (-26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (-0.986 - 4.32i)T + (-27.9 + 13.4i)T^{2} \) |
| 37 | \( 1 + 5.82T + 37T^{2} \) |
| 41 | \( 1 + (-1.69 - 7.43i)T + (-36.9 + 17.7i)T^{2} \) |
| 47 | \( 1 + (7.70 - 3.70i)T + (29.3 - 36.7i)T^{2} \) |
| 53 | \( 1 + (-0.376 + 0.471i)T + (-11.7 - 51.6i)T^{2} \) |
| 59 | \( 1 + (-6.16 + 7.72i)T + (-13.1 - 57.5i)T^{2} \) |
| 61 | \( 1 + (1.65 - 7.23i)T + (-54.9 - 26.4i)T^{2} \) |
| 67 | \( 1 + (7.12 - 3.43i)T + (41.7 - 52.3i)T^{2} \) |
| 71 | \( 1 + (13.8 - 6.68i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (3.11 + 3.90i)T + (-16.2 + 71.1i)T^{2} \) |
| 79 | \( 1 + 2.64T + 79T^{2} \) |
| 83 | \( 1 + (1.79 - 7.85i)T + (-74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (0.629 - 2.75i)T + (-80.1 - 38.6i)T^{2} \) |
| 97 | \( 1 + (-9.72 - 4.68i)T + (60.4 + 75.8i)T^{2} \) |
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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
−11.95353966633823092750383047820, −10.42451240426483734135295046102, −10.04861700399176694642543317386, −9.115401893125582379195935325134, −8.048499404078703249012721541885, −7.09878268247596635465982272048, −6.05190854514205870671918711196, −5.12814172743146395443951689787, −4.06028543971350798459988443555, −3.00555447781153030234686936663,
0.05037569872370564566191189818, 2.10022856630931937744972796731, 3.29554169375489844035028422981, 4.38421828597154459153193367418, 5.93577480875514600359229074213, 6.81139675571369800310971240542, 7.52732487726669206349600309577, 8.997193524611705113016451280082, 9.862565301824893916228287605368, 10.43756185914393212493839977252