| L(s) = 1 | + i·3-s + (−0.539 − 2.17i)5-s − 0.539i·7-s − 9-s − 3.17·11-s + 1.80i·13-s + (2.17 − 0.539i)15-s + 0.290i·17-s − 19-s + 0.539·21-s − 4.78i·23-s + (−4.41 + 2.34i)25-s − i·27-s − 3.17·29-s − 10.0·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (−0.241 − 0.970i)5-s − 0.203i·7-s − 0.333·9-s − 0.955·11-s + 0.499i·13-s + (0.560 − 0.139i)15-s + 0.0705i·17-s − 0.229·19-s + 0.117·21-s − 0.998i·23-s + (−0.883 + 0.468i)25-s − 0.192i·27-s − 0.588·29-s − 1.80·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 + 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1637630977\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1637630977\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (0.539 + 2.17i)T \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + 0.539iT - 7T^{2} \) |
| 11 | \( 1 + 3.17T + 11T^{2} \) |
| 13 | \( 1 - 1.80iT - 13T^{2} \) |
| 17 | \( 1 - 0.290iT - 17T^{2} \) |
| 23 | \( 1 + 4.78iT - 23T^{2} \) |
| 29 | \( 1 + 3.17T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 - 11.2iT - 37T^{2} \) |
| 41 | \( 1 + 8.92T + 41T^{2} \) |
| 43 | \( 1 + 2.53iT - 43T^{2} \) |
| 47 | \( 1 + 4.78iT - 47T^{2} \) |
| 53 | \( 1 - 5.12iT - 53T^{2} \) |
| 59 | \( 1 + 4.52T + 59T^{2} \) |
| 61 | \( 1 + 6.78T + 61T^{2} \) |
| 67 | \( 1 + 8.83iT - 67T^{2} \) |
| 71 | \( 1 - 1.41T + 71T^{2} \) |
| 73 | \( 1 + 9.41iT - 73T^{2} \) |
| 79 | \( 1 + 9.17T + 79T^{2} \) |
| 83 | \( 1 + 6.44iT - 83T^{2} \) |
| 89 | \( 1 + 13.6T + 89T^{2} \) |
| 97 | \( 1 - 18.2iT - 97T^{2} \) |
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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
−9.352902368854841261476590072805, −8.678749494909504393926020523580, −7.975073556564439795056504333034, −7.01449316966241623795989424592, −5.81938083674262702264103509714, −4.99185278016551282751957106291, −4.31785311177062117155907161172, −3.27353151395462252034462980886, −1.83651612764268136192498130247, −0.06629492432569530028924976109,
1.93813679664911382535595653903, 2.92652012788761712284242789498, 3.84389903747570808806704413506, 5.36992839541710575892019365327, 5.91908565591165870917503748992, 7.17404565397424501376035487191, 7.47063828576881443312687221238, 8.384867425987225022823631425790, 9.383090854434051116047011085932, 10.32970794118280769528224951909
