L(s) = 1 | + (0.587 + 0.809i)2-s + (−0.951 + 0.309i)3-s + (−0.309 + 0.951i)4-s + (−1.30 + 1.79i)5-s + (−0.809 − 0.587i)6-s + (2.27 + 1.35i)7-s + (−0.951 + 0.309i)8-s + (0.809 − 0.587i)9-s − 2.21·10-s + (−3.31 + 0.0448i)11-s − i·12-s + (−1.39 + 1.01i)13-s + (0.236 + 2.63i)14-s + (0.684 − 2.10i)15-s + (−0.809 − 0.587i)16-s + (−2.79 − 2.03i)17-s + ⋯ |
L(s) = 1 | + (0.415 + 0.572i)2-s + (−0.549 + 0.178i)3-s + (−0.154 + 0.475i)4-s + (−0.582 + 0.801i)5-s + (−0.330 − 0.239i)6-s + (0.858 + 0.513i)7-s + (−0.336 + 0.109i)8-s + (0.269 − 0.195i)9-s − 0.700·10-s + (−0.999 + 0.0135i)11-s − 0.288i·12-s + (−0.386 + 0.280i)13-s + (0.0630 + 0.704i)14-s + (0.176 − 0.544i)15-s + (−0.202 − 0.146i)16-s + (−0.679 − 0.493i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.141i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 - 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0679089 + 0.956483i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0679089 + 0.956483i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.587 - 0.809i)T \) |
| 3 | \( 1 + (0.951 - 0.309i)T \) |
| 7 | \( 1 + (-2.27 - 1.35i)T \) |
| 11 | \( 1 + (3.31 - 0.0448i)T \) |
good | 5 | \( 1 + (1.30 - 1.79i)T + (-1.54 - 4.75i)T^{2} \) |
| 13 | \( 1 + (1.39 - 1.01i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (2.79 + 2.03i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-1.22 - 3.75i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + 2.14T + 23T^{2} \) |
| 29 | \( 1 + (1.17 + 0.381i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (2.89 + 3.98i)T + (-9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (2.77 - 8.54i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (0.153 + 0.471i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 0.724iT - 43T^{2} \) |
| 47 | \( 1 + (-5.80 + 1.88i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (4.34 - 3.15i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-13.4 - 4.38i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-9.61 - 6.98i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 - 11.8T + 67T^{2} \) |
| 71 | \( 1 + (0.667 + 0.485i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.733 + 2.25i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (1.00 + 1.37i)T + (-24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-6.44 - 4.68i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 - 9.56iT - 89T^{2} \) |
| 97 | \( 1 + (9.02 + 12.4i)T + (-29.9 + 92.2i)T^{2} \) |
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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
−11.53867982006772174364301265249, −10.80788194970437507931707886255, −9.786662669247288608547433804923, −8.489778036232656549288229479679, −7.65126799146940519177098982277, −6.91149100212943675546668895281, −5.71464821600364840904573529470, −4.96379515483883044384610571308, −3.85631020596855685385634421315, −2.43717462534925020219432271554,
0.54156963544111038527578187242, 2.15016571735537589571814445827, 3.90049790071998244262170841734, 4.86072393088897398402594289186, 5.42394902762725261620518561792, 6.95332738694427053460770346734, 7.917637216394967956119133579835, 8.753334207161707335521733707087, 10.04674051731340832823828136036, 10.92895076742647906822291687026