| L(s) = 1 | + (1.41 − 0.0421i)2-s + (1.42 − 0.982i)3-s + (1.99 − 0.119i)4-s + (−3.16 − 2.08i)5-s + (1.97 − 1.44i)6-s + (−1.35 + 1.27i)7-s + (2.81 − 0.252i)8-s + (1.06 − 2.80i)9-s + (−4.56 − 2.80i)10-s + (2.67 − 5.33i)11-s + (2.73 − 2.13i)12-s + (0.0628 + 0.0271i)13-s + (−1.86 + 1.86i)14-s + (−6.56 + 0.141i)15-s + (3.97 − 0.475i)16-s + (−2.02 + 2.41i)17-s + ⋯ |
| L(s) = 1 | + (0.999 − 0.0297i)2-s + (0.823 − 0.567i)3-s + (0.998 − 0.0595i)4-s + (−1.41 − 0.931i)5-s + (0.806 − 0.591i)6-s + (−0.512 + 0.483i)7-s + (0.996 − 0.0892i)8-s + (0.356 − 0.934i)9-s + (−1.44 − 0.888i)10-s + (0.807 − 1.60i)11-s + (0.788 − 0.615i)12-s + (0.0174 + 0.00752i)13-s + (−0.497 + 0.498i)14-s + (−1.69 + 0.0364i)15-s + (0.992 − 0.118i)16-s + (−0.491 + 0.585i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.136 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.136 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.14644 - 1.87169i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.14644 - 1.87169i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.41 + 0.0421i)T \) |
| 3 | \( 1 + (-1.42 + 0.982i)T \) |
| good | 5 | \( 1 + (3.16 + 2.08i)T + (1.98 + 4.59i)T^{2} \) |
| 7 | \( 1 + (1.35 - 1.27i)T + (0.407 - 6.98i)T^{2} \) |
| 11 | \( 1 + (-2.67 + 5.33i)T + (-6.56 - 8.82i)T^{2} \) |
| 13 | \( 1 + (-0.0628 - 0.0271i)T + (8.92 + 9.45i)T^{2} \) |
| 17 | \( 1 + (2.02 - 2.41i)T + (-2.95 - 16.7i)T^{2} \) |
| 19 | \( 1 + (2.74 - 2.29i)T + (3.29 - 18.7i)T^{2} \) |
| 23 | \( 1 + (0.0746 - 0.0791i)T + (-1.33 - 22.9i)T^{2} \) |
| 29 | \( 1 + (-8.41 - 0.983i)T + (28.2 + 6.68i)T^{2} \) |
| 31 | \( 1 + (-0.371 + 1.56i)T + (-27.7 - 13.9i)T^{2} \) |
| 37 | \( 1 + (-5.21 - 0.919i)T + (34.7 + 12.6i)T^{2} \) |
| 41 | \( 1 + (4.52 - 3.37i)T + (11.7 - 39.2i)T^{2} \) |
| 43 | \( 1 + (-0.296 - 5.09i)T + (-42.7 + 4.99i)T^{2} \) |
| 47 | \( 1 + (-7.53 + 1.78i)T + (42.0 - 21.0i)T^{2} \) |
| 53 | \( 1 + (-2.60 + 4.50i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.25 - 8.47i)T + (-35.2 + 47.3i)T^{2} \) |
| 61 | \( 1 + (13.2 + 3.95i)T + (50.9 + 33.5i)T^{2} \) |
| 67 | \( 1 + (3.86 - 0.451i)T + (65.1 - 15.4i)T^{2} \) |
| 71 | \( 1 + (8.54 - 3.11i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (1.21 + 0.440i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-6.57 - 4.89i)T + (22.6 + 75.6i)T^{2} \) |
| 83 | \( 1 + (-4.98 - 3.71i)T + (23.8 + 79.5i)T^{2} \) |
| 89 | \( 1 + (0.658 - 1.80i)T + (-68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-16.0 + 10.5i)T + (38.4 - 89.0i)T^{2} \) |
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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
−10.64452006352525665241536431322, −9.121098479367578702364666852999, −8.420817954228393770129885620667, −7.87771354089775538281845432303, −6.61846861673705313195353148230, −5.97511240443810858211617995447, −4.43658351357675329637906722062, −3.73595699825573460518845718714, −2.87091287801980981611668786558, −1.11858379048690874940402296172,
2.37237882394216044617046829189, 3.32787784173547971348277491225, 4.23854383456010764529679471929, 4.61524567867144847658415170113, 6.61997836625786661395535564888, 7.11375779642153032728111421048, 7.79050877127935137464936688089, 9.019204427006286735616163348423, 10.23015048287698750638304918126, 10.68450414966243589609562437737
