| L(s) = 1 | + (−12.1 − 7.01i)2-s + (54.3 + 60.0i)3-s + (−29.5 − 51.1i)4-s + (676. − 390. i)5-s + (−238. − 1.11e3i)6-s + (2.16e3 − 3.75e3i)7-s + 4.42e3i·8-s + (−656. + 6.52e3i)9-s − 1.09e4·10-s + (5.60e3 + 3.23e3i)11-s + (1.46e3 − 4.55e3i)12-s + (−8.49e3 − 1.47e4i)13-s + (−5.27e4 + 3.04e4i)14-s + (6.01e4 + 1.94e4i)15-s + (2.34e4 − 4.06e4i)16-s + 2.98e4i·17-s + ⋯ |
| L(s) = 1 | + (−0.759 − 0.438i)2-s + (0.670 + 0.741i)3-s + (−0.115 − 0.199i)4-s + (1.08 − 0.624i)5-s + (−0.184 − 0.857i)6-s + (0.903 − 1.56i)7-s + 1.07i·8-s + (−0.100 + 0.994i)9-s − 1.09·10-s + (0.382 + 0.221i)11-s + (0.0707 − 0.219i)12-s + (−0.297 − 0.515i)13-s + (−1.37 + 0.792i)14-s + (1.18 + 0.383i)15-s + (0.358 − 0.620i)16-s + 0.357i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.697 + 0.716i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.697 + 0.716i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.26001 - 0.531527i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.26001 - 0.531527i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-54.3 - 60.0i)T \) |
| good | 2 | \( 1 + (12.1 + 7.01i)T + (128 + 221. i)T^{2} \) |
| 5 | \( 1 + (-676. + 390. i)T + (1.95e5 - 3.38e5i)T^{2} \) |
| 7 | \( 1 + (-2.16e3 + 3.75e3i)T + (-2.88e6 - 4.99e6i)T^{2} \) |
| 11 | \( 1 + (-5.60e3 - 3.23e3i)T + (1.07e8 + 1.85e8i)T^{2} \) |
| 13 | \( 1 + (8.49e3 + 1.47e4i)T + (-4.07e8 + 7.06e8i)T^{2} \) |
| 17 | \( 1 - 2.98e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 9.17e4T + 1.69e10T^{2} \) |
| 23 | \( 1 + (1.06e5 - 6.14e4i)T + (3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 + (-4.34e5 - 2.50e5i)T + (2.50e11 + 4.33e11i)T^{2} \) |
| 31 | \( 1 + (-5.05e5 - 8.76e5i)T + (-4.26e11 + 7.38e11i)T^{2} \) |
| 37 | \( 1 - 1.90e5T + 3.51e12T^{2} \) |
| 41 | \( 1 + (1.95e6 - 1.12e6i)T + (3.99e12 - 6.91e12i)T^{2} \) |
| 43 | \( 1 + (1.64e6 - 2.84e6i)T + (-5.84e12 - 1.01e13i)T^{2} \) |
| 47 | \( 1 + (1.10e5 + 6.36e4i)T + (1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 - 1.40e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (1.79e7 - 1.03e7i)T + (7.34e13 - 1.27e14i)T^{2} \) |
| 61 | \( 1 + (-9.62e6 + 1.66e7i)T + (-9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-1.17e7 - 2.02e7i)T + (-2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 - 9.12e6iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 1.43e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + (-2.69e7 + 4.67e7i)T + (-7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 + (-2.66e7 - 1.53e7i)T + (1.12e15 + 1.95e15i)T^{2} \) |
| 89 | \( 1 + 2.32e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + (-2.46e7 + 4.27e7i)T + (-3.91e15 - 6.78e15i)T^{2} \) |
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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
−19.72177464665583479783750002538, −17.64698644264497059846081039838, −16.91909470157869338371880681419, −14.57248120153992831894796439697, −13.61729355116778299138833117138, −10.68392795167641028963342073623, −9.803628909222628247986358067380, −8.299299819246490366598138892703, −4.78980994530513214802469731579, −1.50946181416601758716779240250,
2.21777943348888205037041890160, 6.41047654190069558581526514836, 8.298717560975629848986144877043, 9.429197454112349421234123163874, 12.11264625731451908589082694814, 13.84764705314494037238445706187, 15.12850466976010735870866576518, 17.32728914712717217232301300687, 18.26166341797226411634486619282, 18.94574482258578317226525433807
