| L(s) = 1 | + (−19.5 − 11.3i)2-s + (−192. − 148. i)3-s + (255. + 443. i)4-s + (−1.21e3 + 698. i)5-s + (2.09e3 + 5.08e3i)6-s + (−1.22e4 + 2.12e4i)7-s − 1.15e4i·8-s + (1.50e4 + 5.71e4i)9-s + 3.16e4·10-s + (8.70e4 + 5.02e4i)11-s + (1.65e4 − 1.23e5i)12-s + (2.48e5 + 4.30e5i)13-s + (4.81e5 − 2.78e5i)14-s + (3.36e5 + 4.50e4i)15-s + (−1.31e5 + 2.27e5i)16-s + 1.31e6i·17-s + ⋯ |
| L(s) = 1 | + (−0.612 − 0.353i)2-s + (−0.792 − 0.610i)3-s + (0.249 + 0.433i)4-s + (−0.387 + 0.223i)5-s + (0.269 + 0.653i)6-s + (−0.731 + 1.26i)7-s − 0.353i·8-s + (0.254 + 0.967i)9-s + 0.316·10-s + (0.540 + 0.312i)11-s + (0.0663 − 0.495i)12-s + (0.670 + 1.16i)13-s + (0.896 − 0.517i)14-s + (0.443 + 0.0593i)15-s + (−0.125 + 0.216i)16-s + 0.928i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 - 0.426i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.904 - 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.121557 + 0.542768i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.121557 + 0.542768i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (19.5 + 11.3i)T \) |
| 3 | \( 1 + (192. + 148. i)T \) |
| 5 | \( 1 + (1.21e3 - 698. i)T \) |
| good | 7 | \( 1 + (1.22e4 - 2.12e4i)T + (-1.41e8 - 2.44e8i)T^{2} \) |
| 11 | \( 1 + (-8.70e4 - 5.02e4i)T + (1.29e10 + 2.24e10i)T^{2} \) |
| 13 | \( 1 + (-2.48e5 - 4.30e5i)T + (-6.89e10 + 1.19e11i)T^{2} \) |
| 17 | \( 1 - 1.31e6iT - 2.01e12T^{2} \) |
| 19 | \( 1 + 3.30e6T + 6.13e12T^{2} \) |
| 23 | \( 1 + (-7.41e5 + 4.27e5i)T + (2.07e13 - 3.58e13i)T^{2} \) |
| 29 | \( 1 + (-2.44e7 - 1.40e7i)T + (2.10e14 + 3.64e14i)T^{2} \) |
| 31 | \( 1 + (3.92e6 + 6.79e6i)T + (-4.09e14 + 7.09e14i)T^{2} \) |
| 37 | \( 1 - 8.06e7T + 4.80e15T^{2} \) |
| 41 | \( 1 + (-4.04e7 + 2.33e7i)T + (6.71e15 - 1.16e16i)T^{2} \) |
| 43 | \( 1 + (9.45e7 - 1.63e8i)T + (-1.08e16 - 1.87e16i)T^{2} \) |
| 47 | \( 1 + (-1.68e7 - 9.72e6i)T + (2.62e16 + 4.55e16i)T^{2} \) |
| 53 | \( 1 - 3.26e8iT - 1.74e17T^{2} \) |
| 59 | \( 1 + (8.59e8 - 4.96e8i)T + (2.55e17 - 4.42e17i)T^{2} \) |
| 61 | \( 1 + (-2.34e8 + 4.05e8i)T + (-3.56e17 - 6.17e17i)T^{2} \) |
| 67 | \( 1 + (-1.23e9 - 2.13e9i)T + (-9.11e17 + 1.57e18i)T^{2} \) |
| 71 | \( 1 - 1.82e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 + 1.52e9T + 4.29e18T^{2} \) |
| 79 | \( 1 + (-2.30e9 + 3.98e9i)T + (-4.73e18 - 8.19e18i)T^{2} \) |
| 83 | \( 1 + (1.55e9 + 8.98e8i)T + (7.75e18 + 1.34e19i)T^{2} \) |
| 89 | \( 1 - 4.51e9iT - 3.11e19T^{2} \) |
| 97 | \( 1 + (6.08e9 - 1.05e10i)T + (-3.68e19 - 6.38e19i)T^{2} \) |
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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
−12.30672767731476042895730992093, −11.54809171553132839688746521848, −10.57148117647246340477553799854, −9.200086601161171358934873940773, −8.257143052639003371321280125148, −6.69510112280157646883674137502, −6.13797808017315632925402673575, −4.24159609192419920010713126802, −2.51749708265255799163785708235, −1.37062029758848305374179246016,
0.28804240191763992405448057016, 0.830372922193959290648223752759, 3.41420758025978381167331923585, 4.55866356456981972225985151692, 6.06407256103799754293540474192, 6.94755131764671839424180374686, 8.310501717229548304613611347649, 9.600788149311799716486731882548, 10.48673191038406146972648266942, 11.23224068129819221286740704524
