L(s) = 1 | + (0.866 + 1.5i)3-s + (−1.73 − i)4-s + (−0.767 − 2.86i)7-s + (−1.5 + 2.59i)9-s − 3.46i·12-s + (−2.59 + 2.5i)13-s + (1.99 + 3.46i)16-s + (7.83 − 2.09i)19-s + (3.63 − 3.63i)21-s + 5i·25-s − 5.19·27-s + (−1.53 + 5.73i)28-s + (−7.83 − 7.83i)31-s + (5.19 − 3i)36-s + (2.09 + 0.562i)37-s + ⋯ |
L(s) = 1 | + (0.499 + 0.866i)3-s + (−0.866 − 0.5i)4-s + (−0.290 − 1.08i)7-s + (−0.5 + 0.866i)9-s − 0.999i·12-s + (−0.720 + 0.693i)13-s + (0.499 + 0.866i)16-s + (1.79 − 0.481i)19-s + (0.792 − 0.792i)21-s + i·25-s − 1.00·27-s + (−0.290 + 1.08i)28-s + (−1.40 − 1.40i)31-s + (0.866 − 0.5i)36-s + (0.344 + 0.0924i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 - 0.277i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 - 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.734633 + 0.103967i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.734633 + 0.103967i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.866 - 1.5i)T \) |
| 13 | \( 1 + (2.59 - 2.5i)T \) |
good | 2 | \( 1 + (1.73 + i)T^{2} \) |
| 5 | \( 1 - 5iT^{2} \) |
| 7 | \( 1 + (0.767 + 2.86i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-7.83 + 2.09i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (7.83 + 7.83i)T + 31iT^{2} \) |
| 37 | \( 1 + (-2.09 - 0.562i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (1.5 + 0.866i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 47iT^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-4.33 + 7.5i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.205 + 0.767i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (9.36 - 9.36i)T - 73iT^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 - 83iT^{2} \) |
| 89 | \( 1 + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (16.4 - 4.40i)T + (84.0 - 48.5i)T^{2} \) |
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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
−16.35775847699612248024532957123, −14.99871328999823903849859673460, −14.04727720450050410999165257189, −13.28427303704086341610784007970, −11.20671029059133093468554275184, −9.878455700632552780346137377328, −9.266331349291400128893358857922, −7.49888675486433736155097520690, −5.18384665668504267031376994747, −3.82871138839055860330868972501,
3.07843900728166971573586321345, 5.50669007436295339404409935203, 7.45618645320222714468884015368, 8.614704473648495175511787933697, 9.662357628454430853550576199998, 12.04244590137973943733356661932, 12.60526945583769593725335756279, 13.81765315447578573887117314647, 14.80180401531769427686703023050, 16.28138119126810073270275984764