L(s) = 1 | − 1.41i·2-s + (1.56 + 2.55i)3-s − 2.00·4-s + 2.23i·5-s + (3.61 − 2.21i)6-s − 1.76·7-s + 2.82i·8-s + (−4.07 + 8.02i)9-s + 3.16·10-s − 17.7i·11-s + (−3.13 − 5.11i)12-s − 20.6·13-s + 2.50i·14-s + (−5.71 + 3.50i)15-s + 4.00·16-s − 0.0288i·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (0.522 + 0.852i)3-s − 0.500·4-s + 0.447i·5-s + (0.602 − 0.369i)6-s − 0.252·7-s + 0.353i·8-s + (−0.453 + 0.891i)9-s + 0.316·10-s − 1.61i·11-s + (−0.261 − 0.426i)12-s − 1.59·13-s + 0.178i·14-s + (−0.381 + 0.233i)15-s + 0.250·16-s − 0.00169i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.852 + 0.522i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.852 + 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5393235615\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5393235615\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 + (-1.56 - 2.55i)T \) |
| 5 | \( 1 - 2.23iT \) |
| 23 | \( 1 + 4.79iT \) |
good | 7 | \( 1 + 1.76T + 49T^{2} \) |
| 11 | \( 1 + 17.7iT - 121T^{2} \) |
| 13 | \( 1 + 20.6T + 169T^{2} \) |
| 17 | \( 1 + 0.0288iT - 289T^{2} \) |
| 19 | \( 1 + 5.91T + 361T^{2} \) |
| 29 | \( 1 + 45.1iT - 841T^{2} \) |
| 31 | \( 1 - 41.1T + 961T^{2} \) |
| 37 | \( 1 + 70.8T + 1.36e3T^{2} \) |
| 41 | \( 1 + 18.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 60.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + 51.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 21.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 68.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 1.38T + 3.72e3T^{2} \) |
| 67 | \( 1 + 100.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 56.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 73.9T + 5.32e3T^{2} \) |
| 79 | \( 1 - 4.47T + 6.24e3T^{2} \) |
| 83 | \( 1 + 21.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 28.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 57.7T + 9.40e3T^{2} \) |
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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
−10.14805230913638784771297244064, −9.181805443067309563914843560310, −8.454839288741212015301162137874, −7.53763824170074585627748411896, −6.14968068641349594918018101438, −5.11562408745000739570515420808, −4.07945639354615004096404858251, −3.10073401944633773982972946621, −2.36987012838892281060198007258, −0.17012780767911316249153850556,
1.61411947762991526969490054730, 2.85688089653935611239891871246, 4.39460857049425293930444881507, 5.20231785770768033273508884428, 6.50660028079987051401679726858, 7.21489196926859769992005379819, 7.75679662564608178785985239441, 8.826069725154963389932916121625, 9.516807041008443854797076549581, 10.25955638532057729148737673516