| L(s) = 1 | + (6.65 − 9.15i)2-s + (32.1 − 99.0i)3-s + (−39.5 − 121. i)4-s + (446. − 324. i)5-s + (−692. − 953. i)6-s + (361. − 117. i)7-s + (−1.37e3 − 447. i)8-s + (−3.46e3 − 2.51e3i)9-s − 6.24e3i·10-s + (−1.02e4 + 1.04e4i)11-s − 1.33e4·12-s + (7.05e3 − 9.71e3i)13-s + (1.32e3 − 4.08e3i)14-s + (−1.77e4 − 5.47e4i)15-s + (−1.32e4 + 9.63e3i)16-s + (−2.46e4 − 3.39e4i)17-s + ⋯ |
| L(s) = 1 | + (0.415 − 0.572i)2-s + (0.397 − 1.22i)3-s + (−0.154 − 0.475i)4-s + (0.714 − 0.519i)5-s + (−0.534 − 0.735i)6-s + (0.150 − 0.0489i)7-s + (−0.336 − 0.109i)8-s + (−0.528 − 0.383i)9-s − 0.624i·10-s + (−0.701 + 0.712i)11-s − 0.642·12-s + (0.247 − 0.340i)13-s + (0.0345 − 0.106i)14-s + (−0.351 − 1.08i)15-s + (−0.202 + 0.146i)16-s + (−0.295 − 0.406i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.882408 - 2.33046i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.882408 - 2.33046i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-6.65 + 9.15i)T \) |
| 11 | \( 1 + (1.02e4 - 1.04e4i)T \) |
| good | 3 | \( 1 + (-32.1 + 99.0i)T + (-5.30e3 - 3.85e3i)T^{2} \) |
| 5 | \( 1 + (-446. + 324. i)T + (1.20e5 - 3.71e5i)T^{2} \) |
| 7 | \( 1 + (-361. + 117. i)T + (4.66e6 - 3.38e6i)T^{2} \) |
| 13 | \( 1 + (-7.05e3 + 9.71e3i)T + (-2.52e8 - 7.75e8i)T^{2} \) |
| 17 | \( 1 + (2.46e4 + 3.39e4i)T + (-2.15e9 + 6.63e9i)T^{2} \) |
| 19 | \( 1 + (3.13e4 + 1.01e4i)T + (1.37e10 + 9.98e9i)T^{2} \) |
| 23 | \( 1 - 3.89e5T + 7.83e10T^{2} \) |
| 29 | \( 1 + (-3.53e5 + 1.14e5i)T + (4.04e11 - 2.94e11i)T^{2} \) |
| 31 | \( 1 + (-1.39e6 - 1.01e6i)T + (2.63e11 + 8.11e11i)T^{2} \) |
| 37 | \( 1 + (-6.40e4 - 1.97e5i)T + (-2.84e12 + 2.06e12i)T^{2} \) |
| 41 | \( 1 + (2.96e6 + 9.61e5i)T + (6.45e12 + 4.69e12i)T^{2} \) |
| 43 | \( 1 + 5.10e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + (-2.32e6 + 7.16e6i)T + (-1.92e13 - 1.39e13i)T^{2} \) |
| 53 | \( 1 + (-7.51e5 - 5.45e5i)T + (1.92e13 + 5.92e13i)T^{2} \) |
| 59 | \( 1 + (-7.38e6 - 2.27e7i)T + (-1.18e14 + 8.63e13i)T^{2} \) |
| 61 | \( 1 + (-1.28e6 - 1.76e6i)T + (-5.92e13 + 1.82e14i)T^{2} \) |
| 67 | \( 1 + 4.20e6T + 4.06e14T^{2} \) |
| 71 | \( 1 + (2.50e6 - 1.81e6i)T + (1.99e14 - 6.14e14i)T^{2} \) |
| 73 | \( 1 + (-2.52e6 + 8.19e5i)T + (6.52e14 - 4.74e14i)T^{2} \) |
| 79 | \( 1 + (3.41e7 - 4.70e7i)T + (-4.68e14 - 1.44e15i)T^{2} \) |
| 83 | \( 1 + (-3.33e7 - 4.58e7i)T + (-6.95e14 + 2.14e15i)T^{2} \) |
| 89 | \( 1 - 2.20e6T + 3.93e15T^{2} \) |
| 97 | \( 1 + (7.09e7 + 5.15e7i)T + (2.42e15 + 7.45e15i)T^{2} \) |
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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
−15.34635692092756401778782519599, −13.73934240377189743519943156217, −13.15407383655183629132326771041, −12.12186566647465522012102224647, −10.30936812575713748847595530256, −8.642322610187129972824787793238, −6.95491725649277211446787028446, −5.10613253658501180551935929561, −2.50597931127391988222568489071, −1.15949007603313831048589052336,
2.99444932576509352438614626596, 4.68648096631771926742944388753, 6.30230195198339455084288615251, 8.406080142486563925676451647325, 9.807576420337134856881797854555, 11.04754827510727608345741744571, 13.21243210431912681800138070203, 14.34430093708215241398972221767, 15.30212576642493123159870394683, 16.29290315716411305555166350749
