| L(s) = 1 | + (0.194 − 0.224i)2-s + (19.9 + 2.87i)3-s + (4.54 + 31.5i)4-s + (−59.2 − 38.0i)5-s + (4.52 − 3.92i)6-s + (−59.5 + 92.6i)7-s + (15.9 + 10.2i)8-s + (158. + 46.4i)9-s + (−20.0 + 5.88i)10-s + (−626. + 402. i)11-s + 644. i·12-s + (86.4 + 12.4i)13-s + (9.20 + 31.3i)14-s + (−1.07e3 − 931. i)15-s + (−974. + 286. i)16-s + (915. + 1.05e3i)17-s + ⋯ |
| L(s) = 1 | + (0.0343 − 0.0396i)2-s + (1.28 + 0.184i)3-s + (0.141 + 0.987i)4-s + (−1.05 − 0.680i)5-s + (0.0513 − 0.0444i)6-s + (−0.459 + 0.714i)7-s + (0.0880 + 0.0566i)8-s + (0.650 + 0.191i)9-s + (−0.0633 + 0.0186i)10-s + (−1.56 + 1.00i)11-s + 1.29i·12-s + (0.141 + 0.0204i)13-s + (0.0125 + 0.0427i)14-s + (−1.23 − 1.06i)15-s + (−0.951 + 0.279i)16-s + (0.768 + 0.886i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.750 - 0.661i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.750 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.496865 + 1.31517i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.496865 + 1.31517i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 89 | \( 1 + (-7.47e4 - 1.51e3i)T \) |
| good | 2 | \( 1 + (-0.194 + 0.224i)T + (-4.55 - 31.6i)T^{2} \) |
| 3 | \( 1 + (-19.9 - 2.87i)T + (233. + 68.4i)T^{2} \) |
| 5 | \( 1 + (59.2 + 38.0i)T + (1.29e3 + 2.84e3i)T^{2} \) |
| 7 | \( 1 + (59.5 - 92.6i)T + (-6.98e3 - 1.52e4i)T^{2} \) |
| 11 | \( 1 + (626. - 402. i)T + (6.69e4 - 1.46e5i)T^{2} \) |
| 13 | \( 1 + (-86.4 - 12.4i)T + (3.56e5 + 1.04e5i)T^{2} \) |
| 17 | \( 1 + (-915. - 1.05e3i)T + (-2.02e5 + 1.40e6i)T^{2} \) |
| 19 | \( 1 + (-382. + 1.30e3i)T + (-2.08e6 - 1.33e6i)T^{2} \) |
| 23 | \( 1 + (-369. + 1.25e3i)T + (-5.41e6 - 3.47e6i)T^{2} \) |
| 29 | \( 1 + (877. - 1.36e3i)T + (-8.52e6 - 1.86e7i)T^{2} \) |
| 31 | \( 1 + (-2.40e3 - 8.17e3i)T + (-2.40e7 + 1.54e7i)T^{2} \) |
| 37 | \( 1 - 1.49e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 + (-4.44e3 + 638. i)T + (1.11e8 - 3.26e7i)T^{2} \) |
| 43 | \( 1 + (3.71e3 + 5.78e3i)T + (-6.10e7 + 1.33e8i)T^{2} \) |
| 47 | \( 1 + (1.17e3 + 8.13e3i)T + (-2.20e8 + 6.46e7i)T^{2} \) |
| 53 | \( 1 + (-3.72e3 + 2.59e4i)T + (-4.01e8 - 1.17e8i)T^{2} \) |
| 59 | \( 1 + (-3.33e4 + 4.79e3i)T + (6.85e8 - 2.01e8i)T^{2} \) |
| 61 | \( 1 + (-1.94e4 - 8.87e3i)T + (5.53e8 + 6.38e8i)T^{2} \) |
| 67 | \( 1 + (7.05e3 - 4.90e4i)T + (-1.29e9 - 3.80e8i)T^{2} \) |
| 71 | \( 1 + (-3.27e3 + 2.10e3i)T + (7.49e8 - 1.64e9i)T^{2} \) |
| 73 | \( 1 + (9.55e3 - 2.80e3i)T + (1.74e9 - 1.12e9i)T^{2} \) |
| 79 | \( 1 + (6.49e4 - 1.90e4i)T + (2.58e9 - 1.66e9i)T^{2} \) |
| 83 | \( 1 + (-5.14e4 + 4.45e4i)T + (5.60e8 - 3.89e9i)T^{2} \) |
| 97 | \( 1 + (-3.40e4 - 2.18e4i)T + (3.56e9 + 7.81e9i)T^{2} \) |
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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
−13.26831688985448309610365252826, −12.65775758090973957990880233312, −11.80813198555949731346010558415, −10.10158796516145534876889387002, −8.634690777710388490795817994271, −8.277960710727601793569877834509, −7.23369162914584030325961249958, −4.84251129235457768616291321262, −3.49115253928079250021626482757, −2.52104824707197467711279509203,
0.45964531105815158079838973992, 2.66288901368292301259382115159, 3.71391967285841883914129741908, 5.70092700841995193555723448512, 7.40996846472072584043563571899, 7.919699788322344655293908190071, 9.488911269840664952068233069584, 10.53472824203158386307690420827, 11.43319607407467265823679055447, 13.22970068619392350535529171879
