| L(s) = 1 | + (−2.57 + 0.546i)2-s + (−1.36 + 2.36i)3-s + (4.48 − 1.99i)4-s + (0.442 − 4.21i)5-s + (2.22 − 6.83i)6-s + (0.115 + 2.64i)7-s + (−6.17 + 4.48i)8-s + (−2.24 − 3.88i)9-s + (1.16 + 11.0i)10-s + (0.0907 + 0.863i)11-s + (−1.40 + 13.3i)12-s + (−0.663 + 2.04i)13-s + (−1.74 − 6.73i)14-s + (9.38 + 6.81i)15-s + (6.85 − 7.61i)16-s + (0.152 + 1.44i)17-s + ⋯ |
| L(s) = 1 | + (−1.81 + 0.386i)2-s + (−0.789 + 1.36i)3-s + (2.24 − 0.997i)4-s + (0.198 − 1.88i)5-s + (0.906 − 2.79i)6-s + (0.0436 + 0.999i)7-s + (−2.18 + 1.58i)8-s + (−0.747 − 1.29i)9-s + (0.368 + 3.50i)10-s + (0.0273 + 0.260i)11-s + (−0.404 + 3.85i)12-s + (−0.184 + 0.566i)13-s + (−0.465 − 1.79i)14-s + (2.42 + 1.75i)15-s + (1.71 − 1.90i)16-s + (0.0369 + 0.351i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00936931 - 0.188010i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00936931 - 0.188010i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-0.115 - 2.64i)T \) |
| 41 | \( 1 + (-2.03 - 6.07i)T \) |
| good | 2 | \( 1 + (2.57 - 0.546i)T + (1.82 - 0.813i)T^{2} \) |
| 3 | \( 1 + (1.36 - 2.36i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.442 + 4.21i)T + (-4.89 - 1.03i)T^{2} \) |
| 11 | \( 1 + (-0.0907 - 0.863i)T + (-10.7 + 2.28i)T^{2} \) |
| 13 | \( 1 + (0.663 - 2.04i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-0.152 - 1.44i)T + (-16.6 + 3.53i)T^{2} \) |
| 19 | \( 1 + (3.42 - 3.80i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (1.49 - 0.318i)T + (21.0 - 9.35i)T^{2} \) |
| 29 | \( 1 + (0.530 + 0.385i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.0651 - 0.620i)T + (-30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (-0.0832 + 0.792i)T + (-36.1 - 7.69i)T^{2} \) |
| 43 | \( 1 + (-0.620 + 1.91i)T + (-34.7 - 25.2i)T^{2} \) |
| 47 | \( 1 + (11.8 - 2.52i)T + (42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (5.41 - 2.41i)T + (35.4 - 39.3i)T^{2} \) |
| 59 | \( 1 + (0.128 + 0.142i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (1.77 - 1.96i)T + (-6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (-8.84 + 3.94i)T + (44.8 - 49.7i)T^{2} \) |
| 71 | \( 1 + (7.99 - 5.80i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (1.63 - 2.82i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.98 - 8.64i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 15.5T + 83T^{2} \) |
| 89 | \( 1 + (-1.26 + 1.40i)T + (-9.30 - 88.5i)T^{2} \) |
| 97 | \( 1 + (-7.06 - 5.13i)T + (29.9 + 92.2i)T^{2} \) |
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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
−11.89167441692408388445101606346, −11.09719781263987835783747902853, −9.843810226506628969251064694703, −9.579578579987804019371380720097, −8.709664477084833909283462994056, −8.114454834071278037543174371733, −6.22718237737950111141626881655, −5.49641377241827352160113832708, −4.44137560786411335056341284068, −1.71276723842776611594730241580,
0.26685247147889636076422685874, 1.93904175448242958706901239135, 3.09194035121651127490782580273, 6.17400613579068893732750031644, 6.86919196508405878016680190658, 7.34794915479744964235183758686, 8.132668329420393497742720495595, 9.753657283020129376000404889999, 10.58073792539039978300583745766, 11.08989153501598751659222518499
