| L(s) = 1 | + (12.6 + 7.31i)2-s + (44.5 + 14.2i)3-s + (42.9 + 74.3i)4-s + (−269. + 466. i)5-s + (459. + 506. i)6-s + (626. − 656. i)7-s − 616. i·8-s + (1.77e3 + 1.27e3i)9-s + (−6.81e3 + 3.93e3i)10-s + (1.64e3 − 949. i)11-s + (848. + 3.92e3i)12-s + 431. i·13-s + (1.27e4 − 3.72e3i)14-s + (−1.86e4 + 1.69e4i)15-s + (1.00e4 − 1.73e4i)16-s + (−1.04e4 − 1.80e4i)17-s + ⋯ |
| L(s) = 1 | + (1.11 + 0.646i)2-s + (0.952 + 0.305i)3-s + (0.335 + 0.580i)4-s + (−0.962 + 1.66i)5-s + (0.868 + 0.957i)6-s + (0.690 − 0.723i)7-s − 0.425i·8-s + (0.813 + 0.582i)9-s + (−2.15 + 1.24i)10-s + (0.372 − 0.215i)11-s + (0.141 + 0.655i)12-s + 0.0544i·13-s + (1.24 − 0.363i)14-s + (−1.42 + 1.29i)15-s + (0.610 − 1.05i)16-s + (−0.513 − 0.889i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.53563 + 2.17555i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.53563 + 2.17555i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-44.5 - 14.2i)T \) |
| 7 | \( 1 + (-626. + 656. i)T \) |
| good | 2 | \( 1 + (-12.6 - 7.31i)T + (64 + 110. i)T^{2} \) |
| 5 | \( 1 + (269. - 466. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 11 | \( 1 + (-1.64e3 + 949. i)T + (9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 - 431. iT - 6.27e7T^{2} \) |
| 17 | \( 1 + (1.04e4 + 1.80e4i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-2.31e4 - 1.33e4i)T + (4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (2.66e4 + 1.54e4i)T + (1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + 4.01e4iT - 1.72e10T^{2} \) |
| 31 | \( 1 + (-2.51e4 + 1.45e4i)T + (1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + (1.43e5 - 2.48e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + 3.09e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 4.73e4T + 2.71e11T^{2} \) |
| 47 | \( 1 + (3.60e5 - 6.23e5i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 + (6.79e5 - 3.92e5i)T + (5.87e11 - 1.01e12i)T^{2} \) |
| 59 | \( 1 + (-4.94e5 - 8.57e5i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-1.30e6 - 7.52e5i)T + (1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (9.97e5 + 1.72e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 - 6.72e5iT - 9.09e12T^{2} \) |
| 73 | \( 1 + (3.19e6 - 1.84e6i)T + (5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-6.26e5 + 1.08e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 - 4.44e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (-7.37e5 + 1.27e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + 8.19e6iT - 8.07e13T^{2} \) |
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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
−16.09285312909232221764460050831, −15.16273960248826803742825009130, −14.30976189734929657960038795257, −13.75983870245142555935690766412, −11.60633835364051546672530374822, −10.17469917554798862273954585185, −7.78876432058503483470600633279, −6.84261511814192051950013368664, −4.34749646250004477780273230632, −3.21838108376547919562405890169,
1.69077114033108899209377448135, 3.81296741911503971831514636373, 5.02514535578734324426968119455, 8.063203549879234357280395927088, 8.938446068959307845094709711522, 11.66955686763352211389779727770, 12.43612033694796943350230998003, 13.35782693691080340458603612681, 14.75670126677566483514185251012, 15.78096025108404627824660259577
