L(s) = 1 | + (−1.40 + 0.114i)2-s + (−1.47 − 0.909i)3-s + (1.97 − 0.321i)4-s + (1 + 2i)5-s + (2.18 + 1.11i)6-s + 7-s + (−2.74 + 0.678i)8-s + (1.34 + 2.68i)9-s + (−1.63 − 2.70i)10-s − 2.94·11-s + (−3.20 − 1.32i)12-s + 1.36i·13-s + (−1.40 + 0.114i)14-s + (0.345 − 3.85i)15-s + (3.79 − 1.26i)16-s − 2.69·17-s + ⋯ |
L(s) = 1 | + (−0.996 + 0.0806i)2-s + (−0.851 − 0.525i)3-s + (0.986 − 0.160i)4-s + (0.447 + 0.894i)5-s + (0.890 + 0.454i)6-s + 0.377·7-s + (−0.970 + 0.239i)8-s + (0.448 + 0.893i)9-s + (−0.517 − 0.855i)10-s − 0.888·11-s + (−0.924 − 0.381i)12-s + 0.378i·13-s + (−0.376 + 0.0304i)14-s + (0.0891 − 0.996i)15-s + (0.948 − 0.317i)16-s − 0.652·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0721 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0721 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.360937 + 0.387995i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.360937 + 0.387995i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.40 - 0.114i)T \) |
| 3 | \( 1 + (1.47 + 0.909i)T \) |
| 5 | \( 1 + (-1 - 2i)T \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + 2.94T + 11T^{2} \) |
| 13 | \( 1 - 1.36iT - 13T^{2} \) |
| 17 | \( 1 + 2.69T + 17T^{2} \) |
| 19 | \( 1 + 3.54iT - 19T^{2} \) |
| 23 | \( 1 - 7.18iT - 23T^{2} \) |
| 29 | \( 1 + 1.36iT - 29T^{2} \) |
| 31 | \( 1 - 8.09iT - 31T^{2} \) |
| 37 | \( 1 - 9.89iT - 37T^{2} \) |
| 41 | \( 1 - 5.89iT - 41T^{2} \) |
| 43 | \( 1 + 2.25T + 43T^{2} \) |
| 47 | \( 1 - 1.81iT - 47T^{2} \) |
| 53 | \( 1 - 13.7T + 53T^{2} \) |
| 59 | \( 1 - 6.80T + 59T^{2} \) |
| 61 | \( 1 + 6.55T + 61T^{2} \) |
| 67 | \( 1 + 8.46T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 8iT - 73T^{2} \) |
| 79 | \( 1 - 16.1iT - 79T^{2} \) |
| 83 | \( 1 + 5.83iT - 83T^{2} \) |
| 89 | \( 1 + 4.83iT - 89T^{2} \) |
| 97 | \( 1 + 3.70iT - 97T^{2} \) |
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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
−11.32070708239260066305121472108, −10.53384377284008460966311283721, −9.875593423251038683379839967755, −8.642361291270927182542916153803, −7.52822686979080204281765577581, −6.93916261513559362323361500961, −6.04606231569944149998662540614, −5.03019503273708777749767764422, −2.83185997903388869420924430454, −1.62336414812565982789499454679,
0.51178340982906139357597110462, 2.21377286195363116400680136314, 4.14272816173143493316226540712, 5.38415341282425124349537645200, 6.09603221673408156285180657106, 7.39235685158977588348194001373, 8.455763810569520325586113589761, 9.175401867935342207652815213508, 10.26263411501852975571021467835, 10.60219009867320951895050422354