| L(s) = 1 | + (−1.30 + 1.14i)3-s + 5-s + (1.35 + 2.27i)7-s + (0.392 − 2.97i)9-s + 5.73i·11-s + 2.24i·13-s + (−1.30 + 1.14i)15-s − 6.50·17-s − 0.217i·19-s + (−4.35 − 1.41i)21-s − 5.57i·23-s + 25-s + (2.88 + 4.32i)27-s + 1.11i·29-s + 5.49i·31-s + ⋯ |
| L(s) = 1 | + (−0.751 + 0.659i)3-s + 0.447·5-s + (0.511 + 0.859i)7-s + (0.130 − 0.991i)9-s + 1.72i·11-s + 0.622i·13-s + (−0.336 + 0.294i)15-s − 1.57·17-s − 0.0499i·19-s + (−0.951 − 0.308i)21-s − 1.16i·23-s + 0.200·25-s + (0.555 + 0.831i)27-s + 0.206i·29-s + 0.987i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 - 0.308i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.951 - 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9896962335\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9896962335\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.30 - 1.14i)T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (-1.35 - 2.27i)T \) |
| good | 11 | \( 1 - 5.73iT - 11T^{2} \) |
| 13 | \( 1 - 2.24iT - 13T^{2} \) |
| 17 | \( 1 + 6.50T + 17T^{2} \) |
| 19 | \( 1 + 0.217iT - 19T^{2} \) |
| 23 | \( 1 + 5.57iT - 23T^{2} \) |
| 29 | \( 1 - 1.11iT - 29T^{2} \) |
| 31 | \( 1 - 5.49iT - 31T^{2} \) |
| 37 | \( 1 - 10.0T + 37T^{2} \) |
| 41 | \( 1 - 1.11T + 41T^{2} \) |
| 43 | \( 1 + 5.89T + 43T^{2} \) |
| 47 | \( 1 + 7.70T + 47T^{2} \) |
| 53 | \( 1 + 8.37iT - 53T^{2} \) |
| 59 | \( 1 - 13.4T + 59T^{2} \) |
| 61 | \( 1 + 1.33iT - 61T^{2} \) |
| 67 | \( 1 + 14.5T + 67T^{2} \) |
| 71 | \( 1 - 7.41iT - 71T^{2} \) |
| 73 | \( 1 - 12.1iT - 73T^{2} \) |
| 79 | \( 1 + 4.54T + 79T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 + 4.45T + 89T^{2} \) |
| 97 | \( 1 + 9.33iT - 97T^{2} \) |
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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
−9.769090829764880389686871042785, −9.043510226195188540906008897476, −8.391234454122367281931020026784, −6.93015297663442837321349934256, −6.58929012302418193922262187691, −5.51515696096770927287620817526, −4.65413632908163716286511907071, −4.33698161893159318870566856973, −2.62318405663078260702561109818, −1.71505392103423138325127844648,
0.41512524943295290211025977981, 1.49458812057934167331201577199, 2.76799902021416702475934695000, 4.04583579805317821282092479251, 5.02960104874623134782610646436, 5.91497149719592314902677116444, 6.39969910581234025419061460439, 7.44407812940380271785011507926, 8.045604262000429347167549333994, 8.864307909150901899776742531056
