L(s) = 1 | + (−0.809 + 0.587i)3-s + (2.04 + 0.909i)5-s + 0.747·7-s + (0.309 − 0.951i)9-s + (0.0646 + 0.198i)11-s + (−0.773 + 2.38i)13-s + (−2.18 + 0.464i)15-s + (5.51 + 4.00i)17-s + (−1.00 − 0.731i)19-s + (−0.604 + 0.439i)21-s + (1.00 + 3.09i)23-s + (3.34 + 3.71i)25-s + (0.309 + 0.951i)27-s + (4.19 − 3.04i)29-s + (−3.02 − 2.19i)31-s + ⋯ |
L(s) = 1 | + (−0.467 + 0.339i)3-s + (0.913 + 0.406i)5-s + 0.282·7-s + (0.103 − 0.317i)9-s + (0.0194 + 0.0599i)11-s + (−0.214 + 0.660i)13-s + (−0.564 + 0.120i)15-s + (1.33 + 0.972i)17-s + (−0.231 − 0.167i)19-s + (−0.131 + 0.0958i)21-s + (0.209 + 0.644i)23-s + (0.669 + 0.743i)25-s + (0.0594 + 0.183i)27-s + (0.778 − 0.565i)29-s + (−0.543 − 0.394i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.699 - 0.714i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.699 - 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.22455 + 0.514754i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.22455 + 0.514754i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.809 - 0.587i)T \) |
| 5 | \( 1 + (-2.04 - 0.909i)T \) |
good | 7 | \( 1 - 0.747T + 7T^{2} \) |
| 11 | \( 1 + (-0.0646 - 0.198i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (0.773 - 2.38i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-5.51 - 4.00i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (1.00 + 0.731i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-1.00 - 3.09i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-4.19 + 3.04i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (3.02 + 2.19i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-0.607 + 1.86i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-0.993 + 3.05i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 12.7T + 43T^{2} \) |
| 47 | \( 1 + (-5.24 + 3.81i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (3.35 - 2.43i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-3.61 + 11.1i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (3.85 + 11.8i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (2.35 + 1.71i)T + (20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (5.29 - 3.85i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (0.778 + 2.39i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-8.28 + 6.02i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-4.59 - 3.33i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (0.284 + 0.876i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-12.5 + 9.13i)T + (29.9 - 92.2i)T^{2} \) |
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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
−11.76302094595952965535139037937, −10.83368620541611594888499315154, −10.02741851376170728570092127554, −9.315040673708826511051237012177, −8.029167243565795665860029948267, −6.78435835930095847033835667391, −5.88650912630169539830950299606, −4.92604053570603960784421293138, −3.48931671844367102238583722718, −1.79397027610451142920268268071,
1.23198642295838561295038337509, 2.87989761038925967401245941719, 4.83003744634610638714514718882, 5.55124303368304231717163739884, 6.61900207027993382267166696958, 7.75356742277304534176730700084, 8.782266349037468392374329405340, 9.917180258399137693532960178877, 10.55864153635753311223659151493, 11.79395889091470212474639251377