| L(s) = 1 | − 3.03i·2-s + (−14.6 + 79.6i)3-s + 246.·4-s + (241. + 44.3i)6-s − 59.7·7-s − 1.52e3i·8-s + (−6.13e3 − 2.32e3i)9-s − 2.18e4i·11-s + (−3.60e3 + 1.96e4i)12-s − 3.84e4·13-s + 181. i·14-s + 5.85e4·16-s − 3.61e4i·17-s + (−7.06e3 + 1.86e4i)18-s − 1.36e5·19-s + ⋯ |
| L(s) = 1 | − 0.189i·2-s + (−0.180 + 0.983i)3-s + 0.964·4-s + (0.186 + 0.0341i)6-s − 0.0248·7-s − 0.372i·8-s + (−0.934 − 0.354i)9-s − 1.49i·11-s + (−0.173 + 0.948i)12-s − 1.34·13-s + 0.00471i·14-s + 0.893·16-s − 0.432i·17-s + (−0.0672 + 0.177i)18-s − 1.04·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.180 + 0.983i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.180 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.712535 - 0.855005i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.712535 - 0.855005i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (14.6 - 79.6i)T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 3.03iT - 256T^{2} \) |
| 7 | \( 1 + 59.7T + 5.76e6T^{2} \) |
| 11 | \( 1 + 2.18e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 3.84e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + 3.61e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 1.36e5T + 1.69e10T^{2} \) |
| 23 | \( 1 - 2.35e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 8.96e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 6.80e4T + 8.52e11T^{2} \) |
| 37 | \( 1 + 6.65e4T + 3.51e12T^{2} \) |
| 41 | \( 1 + 3.76e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 2.03e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + 5.93e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 1.31e7iT - 6.22e13T^{2} \) |
| 59 | \( 1 + 1.58e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 3.51e6T + 1.91e14T^{2} \) |
| 67 | \( 1 - 1.39e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + 1.51e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 3.93e7T + 8.06e14T^{2} \) |
| 79 | \( 1 - 3.15e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + 4.88e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 1.09e8iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 1.16e8T + 7.83e15T^{2} \) |
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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
−12.18648931422777437489181904133, −11.33406373458913527351681269878, −10.49386575766062552354903738433, −9.421504571046534537283012356882, −8.009670542120738984093221919053, −6.45979399691204199735347568083, −5.30522767105418936602559280493, −3.65834405903778420581065174351, −2.45392558045106010515487644735, −0.31441378262736209395400310221,
1.66658202914134662726956193293, 2.59740507405434322160728201938, 4.92018429125416564653388342179, 6.47615971747249099644314671119, 7.16103603460269261798052752679, 8.206124384098000517244709630643, 9.975231787461995141338524849541, 11.14960307779494614647119967625, 12.38395371707385656764254628524, 12.70005831487780786255016690932
