| L(s) = 1 | + 6i·2-s + 27i·3-s + 92·4-s − 162·6-s − 64i·7-s + 1.32e3i·8-s − 729·9-s − 948·11-s + 2.48e3i·12-s + 5.09e3i·13-s + 384·14-s + 3.85e3·16-s + 2.83e4i·17-s − 4.37e3i·18-s + 8.62e3·19-s + ⋯ |
| L(s) = 1 | + 0.530i·2-s + 0.577i·3-s + 0.718·4-s − 0.306·6-s − 0.0705i·7-s + 0.911i·8-s − 0.333·9-s − 0.214·11-s + 0.414i·12-s + 0.643i·13-s + 0.0374·14-s + 0.235·16-s + 1.40i·17-s − 0.176i·18-s + 0.288·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.428700 + 1.81600i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.428700 + 1.81600i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 27iT \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 6iT - 128T^{2} \) |
| 7 | \( 1 + 64iT - 8.23e5T^{2} \) |
| 11 | \( 1 + 948T + 1.94e7T^{2} \) |
| 13 | \( 1 - 5.09e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 - 2.83e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 - 8.62e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 1.52e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 + 3.65e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.76e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.68e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 + 6.29e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 6.85e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 5.83e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 - 4.28e5iT - 1.17e12T^{2} \) |
| 59 | \( 1 + 1.30e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 3.00e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 5.07e5iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 5.56e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.36e6iT - 1.10e13T^{2} \) |
| 79 | \( 1 - 6.91e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 4.37e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 - 8.52e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 8.82e6iT - 8.07e13T^{2} \) |
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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
−13.83730537304705295522036022454, −12.35637536140373273048413582767, −11.21710719538461534120833070799, −10.34620780551102606573043851005, −8.916916967159557663492250418692, −7.70540520776019758486919745325, −6.45278599336597204987675266092, −5.28058417901085053691375727390, −3.62202687984716412447962697966, −1.89559466240585282853950335937,
0.59492147725570881658289728348, 2.10109444820072015065824644526, 3.30969596766132656198520117453, 5.40808952445470138340703521596, 6.83232463280622585182263077114, 7.76473550139272299638928441200, 9.368430594908591086876722559669, 10.64259545087159780556239378648, 11.58635786181038026526055029379, 12.49498826873465102977703193203
