| L(s) = 1 | + 2i·3-s + (1 + 2i)5-s − 9-s − 2i·11-s + (−3 + 2i)13-s + (−4 + 2i)15-s + 6i·19-s + 6i·23-s + (−3 + 4i)25-s + 4i·27-s + 6·29-s − 6i·31-s + 4·33-s − 6·37-s + (−4 − 6i)39-s + ⋯ |
| L(s) = 1 | + 1.15i·3-s + (0.447 + 0.894i)5-s − 0.333·9-s − 0.603i·11-s + (−0.832 + 0.554i)13-s + (−1.03 + 0.516i)15-s + 1.37i·19-s + 1.25i·23-s + (−0.600 + 0.800i)25-s + 0.769i·27-s + 1.11·29-s − 1.07i·31-s + 0.696·33-s − 0.986·37-s + (−0.640 − 0.960i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.868 - 0.496i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.868 - 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.464302170\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.464302170\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1 - 2i)T \) |
| 13 | \( 1 + (3 - 2i)T \) |
| good | 3 | \( 1 - 2iT - 3T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 6iT - 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 6iT - 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 + 8iT - 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 12iT - 53T^{2} \) |
| 59 | \( 1 + 2iT - 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 - 2iT - 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + 8iT - 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−10.04550523434964221350431791893, −9.792937377881083293575322033069, −8.859463484345468218341805707242, −7.75558050714700163873008419714, −6.87765604520897280122427321441, −5.87885084718917339429083425466, −5.10774852424646912255425059314, −3.94275227180622519613045851516, −3.26951814974367224974595473256, −1.94897118320689200813379395203,
0.65121989537002453233270200583, 1.84779343068204927971982666045, 2.82677117847101712530097713933, 4.65102999398781231804203433239, 5.08857877220442595065199326266, 6.52208621393245152975290275549, 6.86546055063972996429447570295, 8.024899303462589330315294464003, 8.512001745118887951581839009846, 9.628517546118631800794722278491
