| L(s) = 1 | − 2-s + 4-s + (2.22 + 0.191i)5-s − 8-s + (−2.22 − 0.191i)10-s − 2.39i·11-s − 5.67·13-s + 16-s + 2.07i·17-s + 5.91i·19-s + (2.22 + 0.191i)20-s + 2.39i·22-s − 1.86·23-s + (4.92 + 0.852i)25-s + 5.67·26-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + (0.996 + 0.0855i)5-s − 0.353·8-s + (−0.704 − 0.0605i)10-s − 0.722i·11-s − 1.57·13-s + 0.250·16-s + 0.502i·17-s + 1.35i·19-s + (0.498 + 0.0427i)20-s + 0.511i·22-s − 0.388·23-s + (0.985 + 0.170i)25-s + 1.11·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.946 - 0.321i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.946 - 0.321i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2821773519\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2821773519\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.22 - 0.191i)T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 2.39iT - 11T^{2} \) |
| 13 | \( 1 + 5.67T + 13T^{2} \) |
| 17 | \( 1 - 2.07iT - 17T^{2} \) |
| 19 | \( 1 - 5.91iT - 19T^{2} \) |
| 23 | \( 1 + 1.86T + 23T^{2} \) |
| 29 | \( 1 - 4.88iT - 29T^{2} \) |
| 31 | \( 1 + 4.52iT - 31T^{2} \) |
| 37 | \( 1 + 2.96iT - 37T^{2} \) |
| 41 | \( 1 + 7.04T + 41T^{2} \) |
| 43 | \( 1 + 8.55iT - 43T^{2} \) |
| 47 | \( 1 - 5.57iT - 47T^{2} \) |
| 53 | \( 1 + 4.19T + 53T^{2} \) |
| 59 | \( 1 - 2.00T + 59T^{2} \) |
| 61 | \( 1 + 12.4iT - 61T^{2} \) |
| 67 | \( 1 - 7.62iT - 67T^{2} \) |
| 71 | \( 1 - 9.14iT - 71T^{2} \) |
| 73 | \( 1 + 1.08T + 73T^{2} \) |
| 79 | \( 1 + 16.7T + 79T^{2} \) |
| 83 | \( 1 - 13.6iT - 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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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
−8.625313795535635979317297749209, −8.114087325612472545515853232715, −7.23218775041546372385166326962, −6.61456583420046094271246479714, −5.71248453123683477864989919851, −5.33989568032817280529234395264, −4.09831499534222819398067674477, −3.05565970018395898267166739396, −2.20769571523337354722157575684, −1.40892753140194695367390469377,
0.091693106533904806507233147203, 1.49467429421429288332896521971, 2.40353739444031430094124750386, 2.95033785105341213219736126103, 4.57174150870531663848752634400, 4.99651830752402667440851284955, 5.93009383953498991948457671387, 6.87527124921539466236611641668, 7.15387231778544138410170186564, 8.080084828259274768443711837211
