| L(s) = 1 | + (−980. + 566. i)5-s + (1.54e3 − 2.67e3i)7-s + (980. + 566. i)11-s + (3.64e3 + 6.31e3i)13-s + 5.89e4i·17-s − 8.03e4·19-s + (−8.43e4 + 4.87e4i)23-s + (4.46e5 − 7.72e5i)25-s + (7.48e5 + 4.32e5i)29-s + (−2.17e5 − 3.77e5i)31-s + 3.50e6i·35-s + 1.15e6·37-s + (−2.35e6 + 1.35e6i)41-s + (−4.95e5 + 8.57e5i)43-s + (5.80e6 + 3.35e6i)47-s + ⋯ |
| L(s) = 1 | + (−1.56 + 0.906i)5-s + (0.644 − 1.11i)7-s + (0.0670 + 0.0386i)11-s + (0.127 + 0.221i)13-s + 0.705i·17-s − 0.616·19-s + (−0.301 + 0.174i)23-s + (1.14 − 1.97i)25-s + (1.05 + 0.610i)29-s + (−0.236 − 0.408i)31-s + 2.33i·35-s + 0.618·37-s + (−0.832 + 0.480i)41-s + (−0.144 + 0.250i)43-s + (1.18 + 0.686i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.3182181139\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3182181139\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (980. - 566. i)T + (1.95e5 - 3.38e5i)T^{2} \) |
| 7 | \( 1 + (-1.54e3 + 2.67e3i)T + (-2.88e6 - 4.99e6i)T^{2} \) |
| 11 | \( 1 + (-980. - 566. i)T + (1.07e8 + 1.85e8i)T^{2} \) |
| 13 | \( 1 + (-3.64e3 - 6.31e3i)T + (-4.07e8 + 7.06e8i)T^{2} \) |
| 17 | \( 1 - 5.89e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 8.03e4T + 1.69e10T^{2} \) |
| 23 | \( 1 + (8.43e4 - 4.87e4i)T + (3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 + (-7.48e5 - 4.32e5i)T + (2.50e11 + 4.33e11i)T^{2} \) |
| 31 | \( 1 + (2.17e5 + 3.77e5i)T + (-4.26e11 + 7.38e11i)T^{2} \) |
| 37 | \( 1 - 1.15e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + (2.35e6 - 1.35e6i)T + (3.99e12 - 6.91e12i)T^{2} \) |
| 43 | \( 1 + (4.95e5 - 8.57e5i)T + (-5.84e12 - 1.01e13i)T^{2} \) |
| 47 | \( 1 + (-5.80e6 - 3.35e6i)T + (1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 - 1.00e7iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (-1.38e6 + 7.99e5i)T + (7.34e13 - 1.27e14i)T^{2} \) |
| 61 | \( 1 + (9.68e6 - 1.67e7i)T + (-9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-1.40e7 - 2.42e7i)T + (-2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 + 3.36e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 2.52e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + (-3.17e7 + 5.49e7i)T + (-7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 + (4.11e7 + 2.37e7i)T + (1.12e15 + 1.95e15i)T^{2} \) |
| 89 | \( 1 + 7.82e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + (9.77e6 - 1.69e7i)T + (-3.91e15 - 6.78e15i)T^{2} \) |
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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
−10.31582293240025704686560383782, −8.678287061796805546243948405185, −7.81009913586690114673068511395, −7.24530518010861372501426285287, −6.27135937933247699979368148931, −4.46726885611998399805418049954, −4.01335631690749118353639656327, −2.90853848955852167003406593307, −1.30114900938714044914439889994, −0.082460914062800496435394924326,
0.924070345090790133409683396140, 2.34219289209548342589335809049, 3.68960808543985377799557770690, 4.68287513344033440161377303150, 5.43026579478538334181023058169, 6.87930607289050590409246193557, 8.146469048662545949180545386406, 8.391542997547529727329145217825, 9.365155393731117511918114021805, 10.80336085884998647090786744545
