L(s) = 1 | + (80.1 − 80.1i)3-s + (621. + 62.6i)5-s + (−1.04e3 − 1.04e3i)7-s − 6.27e3i·9-s − 1.09e3·11-s + (3.65e4 − 3.65e4i)13-s + (5.48e4 − 4.48e4i)15-s + (−6.72e4 − 6.72e4i)17-s + 1.77e5i·19-s − 1.67e5·21-s + (−1.10e5 + 1.10e5i)23-s + (3.82e5 + 7.79e4i)25-s + (2.25e4 + 2.25e4i)27-s + 9.03e5i·29-s − 1.11e6·31-s + ⋯ |
L(s) = 1 | + (0.989 − 0.989i)3-s + (0.994 + 0.100i)5-s + (−0.435 − 0.435i)7-s − 0.957i·9-s − 0.0749·11-s + (1.28 − 1.28i)13-s + (1.08 − 0.885i)15-s + (−0.804 − 0.804i)17-s + 1.36i·19-s − 0.861·21-s + (−0.396 + 0.396i)23-s + (0.979 + 0.199i)25-s + (0.0423 + 0.0423i)27-s + 1.27i·29-s − 1.21·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.437 + 0.899i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(2.12383 - 1.32808i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.12383 - 1.32808i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-621. - 62.6i)T \) |
good | 3 | \( 1 + (-80.1 + 80.1i)T - 6.56e3iT^{2} \) |
| 7 | \( 1 + (1.04e3 + 1.04e3i)T + 5.76e6iT^{2} \) |
| 11 | \( 1 + 1.09e3T + 2.14e8T^{2} \) |
| 13 | \( 1 + (-3.65e4 + 3.65e4i)T - 8.15e8iT^{2} \) |
| 17 | \( 1 + (6.72e4 + 6.72e4i)T + 6.97e9iT^{2} \) |
| 19 | \( 1 - 1.77e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 + (1.10e5 - 1.10e5i)T - 7.83e10iT^{2} \) |
| 29 | \( 1 - 9.03e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 1.11e6T + 8.52e11T^{2} \) |
| 37 | \( 1 + (-1.48e6 - 1.48e6i)T + 3.51e12iT^{2} \) |
| 41 | \( 1 - 4.18e6T + 7.98e12T^{2} \) |
| 43 | \( 1 + (1.23e6 - 1.23e6i)T - 1.16e13iT^{2} \) |
| 47 | \( 1 + (-3.39e6 - 3.39e6i)T + 2.38e13iT^{2} \) |
| 53 | \( 1 + (6.26e6 - 6.26e6i)T - 6.22e13iT^{2} \) |
| 59 | \( 1 - 4.61e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 1.06e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + (3.39e6 + 3.39e6i)T + 4.06e14iT^{2} \) |
| 71 | \( 1 + 4.31e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + (9.30e6 - 9.30e6i)T - 8.06e14iT^{2} \) |
| 79 | \( 1 + 5.42e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 + (-4.43e7 + 4.43e7i)T - 2.25e15iT^{2} \) |
| 89 | \( 1 + 2.12e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + (4.71e7 + 4.71e7i)T + 7.83e15iT^{2} \) |
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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
−16.23015124672337795935200886529, −14.50568193322817963745611304381, −13.48116839905106484510619103884, −12.84256562951720178488936848485, −10.57682714010000478848168616280, −9.023269743451955318866213765715, −7.59316665787641425230654589331, −6.04940699757793439287152211671, −3.08519779852104617380974202390, −1.40176512064521413425336520465,
2.31964632469469384098061149790, 4.15715734147188816438990658587, 6.23750678637450605036127611333, 8.817924298035328812887307336259, 9.409896351362601471332231624102, 10.92942774413343907723118504140, 13.12744721099551365821635579861, 14.12904698428743573325271427026, 15.38477168648855534073922655044, 16.38782134657297832892327183769