L(s) = 1 | + (1.14 + 0.827i)2-s + (1.73 + 0.0404i)3-s + (0.630 + 1.89i)4-s + (0.0573 + 0.0573i)5-s + (1.95 + 1.47i)6-s − 7-s + (−0.847 + 2.69i)8-s + (2.99 + 0.139i)9-s + (0.0183 + 0.113i)10-s + (1.96 − 1.96i)11-s + (1.01 + 3.31i)12-s + (−4.48 − 4.48i)13-s + (−1.14 − 0.827i)14-s + (0.0969 + 0.101i)15-s + (−3.20 + 2.39i)16-s + 4.71i·17-s + ⋯ |
L(s) = 1 | + (0.810 + 0.585i)2-s + (0.999 + 0.0233i)3-s + (0.315 + 0.949i)4-s + (0.0256 + 0.0256i)5-s + (0.797 + 0.603i)6-s − 0.377·7-s + (−0.299 + 0.954i)8-s + (0.998 + 0.0466i)9-s + (0.00579 + 0.0358i)10-s + (0.592 − 0.592i)11-s + (0.292 + 0.956i)12-s + (−1.24 − 1.24i)13-s + (−0.306 − 0.221i)14-s + (0.0250 + 0.0262i)15-s + (−0.801 + 0.598i)16-s + 1.14i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.531 - 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.531 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.29556 + 1.27005i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.29556 + 1.27005i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.14 - 0.827i)T \) |
| 3 | \( 1 + (-1.73 - 0.0404i)T \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + (-0.0573 - 0.0573i)T + 5iT^{2} \) |
| 11 | \( 1 + (-1.96 + 1.96i)T - 11iT^{2} \) |
| 13 | \( 1 + (4.48 + 4.48i)T + 13iT^{2} \) |
| 17 | \( 1 - 4.71iT - 17T^{2} \) |
| 19 | \( 1 + (2.60 - 2.60i)T - 19iT^{2} \) |
| 23 | \( 1 + 2.30iT - 23T^{2} \) |
| 29 | \( 1 + (-3.24 + 3.24i)T - 29iT^{2} \) |
| 31 | \( 1 + 1.26iT - 31T^{2} \) |
| 37 | \( 1 + (2.13 - 2.13i)T - 37iT^{2} \) |
| 41 | \( 1 + 6.69T + 41T^{2} \) |
| 43 | \( 1 + (-1.86 - 1.86i)T + 43iT^{2} \) |
| 47 | \( 1 - 5.14T + 47T^{2} \) |
| 53 | \( 1 + (-1.91 - 1.91i)T + 53iT^{2} \) |
| 59 | \( 1 + (3.22 - 3.22i)T - 59iT^{2} \) |
| 61 | \( 1 + (7.78 + 7.78i)T + 61iT^{2} \) |
| 67 | \( 1 + (-8.73 + 8.73i)T - 67iT^{2} \) |
| 71 | \( 1 - 12.9iT - 71T^{2} \) |
| 73 | \( 1 + 8.63iT - 73T^{2} \) |
| 79 | \( 1 - 13.8iT - 79T^{2} \) |
| 83 | \( 1 + (-7.95 - 7.95i)T + 83iT^{2} \) |
| 89 | \( 1 + 18.1T + 89T^{2} \) |
| 97 | \( 1 - 4.55T + 97T^{2} \) |
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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
−12.33628781230324318190673978236, −10.68587484056800598784281103207, −9.805271461444573790825102107020, −8.458474014375969302708674426350, −8.036790744199247102358206169231, −6.84177332315141347137478548292, −5.92281788801028149503912564152, −4.51955135097874497524039703723, −3.50962024336141247988388726470, −2.44532899536332371499184392052,
1.85049465269201974588259815943, 2.94596221956239586963221756759, 4.17569413789645157772136952706, 5.03228691580670597060185499778, 6.81853490806442223896532000480, 7.20209374702313886245453435227, 9.111582485646285476299382791449, 9.432568021938645920802031075614, 10.42685698449122393023462321065, 11.72531699838810701437531671094