L(s) = 1 | − 2-s + 4-s + (1.88 + 1.20i)5-s + (2.61 − 0.384i)7-s − 8-s + (−1.88 − 1.20i)10-s + 0.149i·11-s − 5.72·13-s + (−2.61 + 0.384i)14-s + 16-s + 6.33i·17-s − 5.66i·19-s + (1.88 + 1.20i)20-s − 0.149i·22-s + 4.59·23-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + (0.843 + 0.537i)5-s + (0.989 − 0.145i)7-s − 0.353·8-s + (−0.596 − 0.380i)10-s + 0.0450i·11-s − 1.58·13-s + (−0.699 + 0.102i)14-s + 0.250·16-s + 1.53i·17-s − 1.29i·19-s + (0.421 + 0.268i)20-s − 0.0318i·22-s + 0.958·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.409 - 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.409 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.457540884\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.457540884\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.88 - 1.20i)T \) |
| 7 | \( 1 + (-2.61 + 0.384i)T \) |
good | 11 | \( 1 - 0.149iT - 11T^{2} \) |
| 13 | \( 1 + 5.72T + 13T^{2} \) |
| 17 | \( 1 - 6.33iT - 17T^{2} \) |
| 19 | \( 1 + 5.66iT - 19T^{2} \) |
| 23 | \( 1 - 4.59T + 23T^{2} \) |
| 29 | \( 1 - 5.06iT - 29T^{2} \) |
| 31 | \( 1 - 7.39iT - 31T^{2} \) |
| 37 | \( 1 - 5.76iT - 37T^{2} \) |
| 41 | \( 1 - 4.02T + 41T^{2} \) |
| 43 | \( 1 + 4.14iT - 43T^{2} \) |
| 47 | \( 1 - 12.0iT - 47T^{2} \) |
| 53 | \( 1 + 9.38T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 15.6iT - 67T^{2} \) |
| 71 | \( 1 + 3.23iT - 71T^{2} \) |
| 73 | \( 1 - 8.74T + 73T^{2} \) |
| 79 | \( 1 + 2.05T + 79T^{2} \) |
| 83 | \( 1 - 8.07iT - 83T^{2} \) |
| 89 | \( 1 + 2.31T + 89T^{2} \) |
| 97 | \( 1 + 6.70T + 97T^{2} \) |
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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
−9.347943149184431204133724292548, −8.687854850851191666497782212062, −7.78879205831061606387097824998, −7.05201896594628777462069090214, −6.45940672321723546143576561586, −5.27155451967824429458776043601, −4.71293009501750444816795638524, −3.14984410317298479009842707277, −2.25232006847972730626391680136, −1.32171992270210061649800162859,
0.70028471686178530422350662437, 2.00180362495389470320944461895, 2.62777606610457330225333600299, 4.30615682811504977604891089247, 5.19843924105326764353610703097, 5.72148009093501038623850646767, 6.94418366441974908832368180353, 7.63353149246292155807564122276, 8.307723410486560855277438625085, 9.241614316768414593460052288588