L(s) = 1 | + (−1.37 + 1.04i)3-s + (1.84 + 1.26i)5-s + (2.63 − 0.269i)7-s + (0.801 − 2.89i)9-s − 3.01i·11-s + 4.39·13-s + (−3.86 + 0.180i)15-s − 2.71i·17-s − 8.23i·19-s + (−3.34 + 3.13i)21-s − 3.81·23-s + (1.77 + 4.67i)25-s + (1.92 + 4.82i)27-s + 1.17i·29-s + 1.73i·31-s + ⋯ |
L(s) = 1 | + (−0.796 + 0.605i)3-s + (0.823 + 0.567i)5-s + (0.994 − 0.101i)7-s + (0.267 − 0.963i)9-s − 0.908i·11-s + 1.21·13-s + (−0.998 + 0.0465i)15-s − 0.657i·17-s − 1.88i·19-s + (−0.730 + 0.683i)21-s − 0.796·23-s + (0.355 + 0.934i)25-s + (0.370 + 0.928i)27-s + 0.217i·29-s + 0.311i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.147i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.60405 + 0.119335i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.60405 + 0.119335i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.37 - 1.04i)T \) |
| 5 | \( 1 + (-1.84 - 1.26i)T \) |
| 7 | \( 1 + (-2.63 + 0.269i)T \) |
good | 11 | \( 1 + 3.01iT - 11T^{2} \) |
| 13 | \( 1 - 4.39T + 13T^{2} \) |
| 17 | \( 1 + 2.71iT - 17T^{2} \) |
| 19 | \( 1 + 8.23iT - 19T^{2} \) |
| 23 | \( 1 + 3.81T + 23T^{2} \) |
| 29 | \( 1 - 1.17iT - 29T^{2} \) |
| 31 | \( 1 - 1.73iT - 31T^{2} \) |
| 37 | \( 1 + 4.60iT - 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 + 9.18iT - 43T^{2} \) |
| 47 | \( 1 - 12.0iT - 47T^{2} \) |
| 53 | \( 1 + 7.14T + 53T^{2} \) |
| 59 | \( 1 + 9.11T + 59T^{2} \) |
| 61 | \( 1 - 13.5iT - 61T^{2} \) |
| 67 | \( 1 + 0.494iT - 67T^{2} \) |
| 71 | \( 1 - 5.15iT - 71T^{2} \) |
| 73 | \( 1 - 4.76T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 - 8.16iT - 83T^{2} \) |
| 89 | \( 1 - 2.26T + 89T^{2} \) |
| 97 | \( 1 - 2.92T + 97T^{2} \) |
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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
−10.54221137263759219229033033724, −9.315046852310517249044894078500, −8.859647469167176167519247740265, −7.54651808584900235911626286803, −6.51886315215911106931402105498, −5.80636791438051368113662384606, −5.02220679690607351791989538080, −3.96112611112404232851578566006, −2.70014062898935119003880464941, −1.05081114701054797334599831026,
1.40843318152234158714390859554, 1.95834392838596182335081563989, 4.07403975175403529306029857837, 5.00002415688926045760669542794, 5.93024033606109886407191303132, 6.37843904973845356856109836049, 7.927405174439317531560060322014, 8.117503488548110026148465728405, 9.439703779958683951274026317066, 10.31963893662486122965858199327