| L(s) = 1 | + (3 − 26.8i)3-s + 160. i·5-s − 242·7-s + (−711. − 160. i)9-s + 1.77e3i·11-s + 2.61e3·13-s + (4.32e3 + 482. i)15-s + 7.08e3i·17-s − 5.78e3·19-s + (−726 + 6.49e3i)21-s + 9.33e3i·23-s − 1.02e4·25-s + (−6.45e3 + 1.85e4i)27-s − 1.23e4i·29-s + 2.04e4·31-s + ⋯ |
| L(s) = 1 | + (0.111 − 0.993i)3-s + 1.28i·5-s − 0.705·7-s + (−0.975 − 0.220i)9-s + 1.33i·11-s + 1.19·13-s + (1.28 + 0.143i)15-s + 1.44i·17-s − 0.843·19-s + (−0.0783 + 0.701i)21-s + 0.767i·23-s − 0.658·25-s + (−0.327 + 0.944i)27-s − 0.508i·29-s + 0.686·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.111 - 0.993i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.111 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.886010 + 0.792471i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.886010 + 0.792471i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-3 + 26.8i)T \) |
| good | 5 | \( 1 - 160. iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 242T + 1.17e5T^{2} \) |
| 11 | \( 1 - 1.77e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 2.61e3T + 4.82e6T^{2} \) |
| 17 | \( 1 - 7.08e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 5.78e3T + 4.70e7T^{2} \) |
| 23 | \( 1 - 9.33e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 1.23e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 2.04e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 4.67e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 3.54e3iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 6.86e4T + 6.32e9T^{2} \) |
| 47 | \( 1 - 2.12e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 1.71e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 1.49e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 2.47e4T + 5.15e10T^{2} \) |
| 67 | \( 1 - 8.43e4T + 9.04e10T^{2} \) |
| 71 | \( 1 - 3.24e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 1.13e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 1.59e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 5.15e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 1.25e6iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 8.99e5T + 8.32e11T^{2} \) |
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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
−14.65266381376771092999303965711, −13.42393757641612420097710587402, −12.53103406375134193489638390999, −11.17304685344180160947405309743, −10.02082547999777236166588642992, −8.300158916000469484923674920957, −6.91443005779485860572243554041, −6.21302865394191111538034271369, −3.50736264815613614278849101604, −1.92859426622313922842626802216,
0.52419967384380429525887302467, 3.29622682910930402621282178445, 4.76899565129540580136146036217, 6.08990079236854987198885290601, 8.527421664979238672122677835970, 9.040126814414842303447432392661, 10.47761928837802605962565675153, 11.69631460272486440108337641524, 13.12886861110834786905501853432, 14.02549310874133171055871603873
