| L(s) = 1 | − 14.5i·2-s + 43.4·4-s + 870.·5-s − 4.07e3i·7-s − 4.36e3i·8-s − 1.26e4i·10-s + (1.07e4 − 9.94e3i)11-s + 3.71e4i·13-s − 5.94e4·14-s − 5.25e4·16-s − 2.00e3i·17-s − 8.95e4i·19-s + 3.78e4·20-s + (−1.44e5 − 1.56e5i)22-s + 2.77e5·23-s + ⋯ |
| L(s) = 1 | − 0.911i·2-s + 0.169·4-s + 1.39·5-s − 1.69i·7-s − 1.06i·8-s − 1.26i·10-s + (0.734 − 0.679i)11-s + 1.29i·13-s − 1.54·14-s − 0.801·16-s − 0.0239i·17-s − 0.686i·19-s + 0.236·20-s + (−0.618 − 0.668i)22-s + 0.991·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.734 + 0.679i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.734 + 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.17000 - 2.98813i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.17000 - 2.98813i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (-1.07e4 + 9.94e3i)T \) |
| good | 2 | \( 1 + 14.5iT - 256T^{2} \) |
| 5 | \( 1 - 870.T + 3.90e5T^{2} \) |
| 7 | \( 1 + 4.07e3iT - 5.76e6T^{2} \) |
| 13 | \( 1 - 3.71e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + 2.00e3iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 8.95e4iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 2.77e5T + 7.83e10T^{2} \) |
| 29 | \( 1 - 3.25e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 4.44e5T + 8.52e11T^{2} \) |
| 37 | \( 1 + 1.48e6T + 3.51e12T^{2} \) |
| 41 | \( 1 - 1.84e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 3.67e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + 3.89e6T + 2.38e13T^{2} \) |
| 53 | \( 1 - 6.99e6T + 6.22e13T^{2} \) |
| 59 | \( 1 - 4.71e6T + 1.46e14T^{2} \) |
| 61 | \( 1 + 7.61e6iT - 1.91e14T^{2} \) |
| 67 | \( 1 + 1.80e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + 4.47e6T + 6.45e14T^{2} \) |
| 73 | \( 1 - 1.06e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 + 4.97e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 + 6.38e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 1.01e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + 1.28e8T + 7.83e15T^{2} \) |
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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
−11.60092164781385075318535719148, −10.82351372757761911721038739892, −9.944993091021456841343648734977, −9.091886377470389108545928165433, −7.04159608087420884538579477258, −6.40321135869583151329936099388, −4.50102246356379774366585017404, −3.21478796037658261159979614970, −1.71502653057883163241973697329, −0.917956235821705968884873063862,
1.76080489384544735113173241391, 2.71050161091133845198043980041, 5.32114651511740927078072723390, 5.77935345543696829891306812546, 6.82273925341447463614359371851, 8.323401425096496482634023610927, 9.238099140431217054683070163791, 10.34209165488236186277411005704, 11.82399651328201115441416959545, 12.71732506944084370064933263246
