| L(s) = 1 | + (−0.911 − 0.411i)2-s + (0.591 − 1.26i)3-s + (0.660 + 0.750i)4-s + (1.89 − 0.435i)5-s + (−1.05 + 0.904i)6-s + (0.489 + 0.794i)7-s + (−0.292 − 0.956i)8-s + (0.678 + 0.816i)9-s + (−1.90 − 0.381i)10-s + (0.284 + 1.52i)11-s + (1.33 − 0.388i)12-s + (4.40 + 0.374i)13-s + (−0.118 − 0.925i)14-s + (0.569 − 2.64i)15-s + (−0.127 + 0.991i)16-s + (5.05 + 1.78i)17-s + ⋯ |
| L(s) = 1 | + (−0.644 − 0.291i)2-s + (0.341 − 0.727i)3-s + (0.330 + 0.375i)4-s + (0.845 − 0.194i)5-s + (−0.431 + 0.369i)6-s + (0.184 + 0.300i)7-s + (−0.103 − 0.338i)8-s + (0.226 + 0.272i)9-s + (−0.601 − 0.120i)10-s + (0.0857 + 0.460i)11-s + (0.385 − 0.112i)12-s + (1.22 + 0.104i)13-s + (−0.0316 − 0.247i)14-s + (0.146 − 0.681i)15-s + (−0.0317 + 0.247i)16-s + (1.22 + 0.433i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 446 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 + 0.647i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 446 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.762 + 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.35608 - 0.498240i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.35608 - 0.498240i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.911 + 0.411i)T \) |
| 223 | \( 1 + (-12.3 + 8.36i)T \) |
| good | 3 | \( 1 + (-0.591 + 1.26i)T + (-1.91 - 2.30i)T^{2} \) |
| 5 | \( 1 + (-1.89 + 0.435i)T + (4.49 - 2.18i)T^{2} \) |
| 7 | \( 1 + (-0.489 - 0.794i)T + (-3.15 + 6.25i)T^{2} \) |
| 11 | \( 1 + (-0.284 - 1.52i)T + (-10.2 + 3.95i)T^{2} \) |
| 13 | \( 1 + (-4.40 - 0.374i)T + (12.8 + 2.19i)T^{2} \) |
| 17 | \( 1 + (-5.05 - 1.78i)T + (13.2 + 10.6i)T^{2} \) |
| 19 | \( 1 + (5.64 + 0.320i)T + (18.8 + 2.14i)T^{2} \) |
| 23 | \( 1 + (3.42 + 3.67i)T + (-1.62 + 22.9i)T^{2} \) |
| 29 | \( 1 + (0.106 - 0.0446i)T + (20.3 - 20.6i)T^{2} \) |
| 31 | \( 1 + (-1.18 - 7.57i)T + (-29.5 + 9.49i)T^{2} \) |
| 37 | \( 1 + (6.56 + 5.61i)T + (5.73 + 36.5i)T^{2} \) |
| 41 | \( 1 + (0.442 + 10.4i)T + (-40.8 + 3.47i)T^{2} \) |
| 43 | \( 1 + (0.497 + 7.01i)T + (-42.5 + 6.06i)T^{2} \) |
| 47 | \( 1 + (-4.50 + 0.255i)T + (46.6 - 5.30i)T^{2} \) |
| 53 | \( 1 + (-1.31 - 0.786i)T + (25.1 + 46.6i)T^{2} \) |
| 59 | \( 1 + (2.65 - 2.54i)T + (2.50 - 58.9i)T^{2} \) |
| 61 | \( 1 + (5.06 - 7.27i)T + (-21.1 - 57.2i)T^{2} \) |
| 67 | \( 1 + (7.33 - 1.47i)T + (61.8 - 25.8i)T^{2} \) |
| 71 | \( 1 + (2.41 + 9.83i)T + (-62.9 + 32.8i)T^{2} \) |
| 73 | \( 1 + (-1.92 + 4.41i)T + (-49.7 - 53.4i)T^{2} \) |
| 79 | \( 1 + (7.44 - 4.72i)T + (33.5 - 71.5i)T^{2} \) |
| 83 | \( 1 + (1.03 - 14.6i)T + (-82.1 - 11.7i)T^{2} \) |
| 89 | \( 1 + (-7.03 - 1.61i)T + (80.0 + 38.9i)T^{2} \) |
| 97 | \( 1 + (11.0 - 11.8i)T + (-6.85 - 96.7i)T^{2} \) |
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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
−10.52242790002268574091144501967, −10.39375475018052351287477868595, −8.932581265053135405729696569412, −8.511704993962485547577804663156, −7.45623707074519534045083120919, −6.49117312498102814294119595457, −5.49907750788171885522732770871, −3.88982554185930520417761847851, −2.24458088714567656059728146103, −1.50113649669888788252737400339,
1.43969744070462915358865600498, 3.15106643956856246426903780096, 4.31843374866196807740718206696, 5.81825051521885727515502990741, 6.39366823131370504024263878307, 7.77552049675736609975815063196, 8.588278956546965882143909989785, 9.563874208053885868823492290489, 10.05956041300194297247155965712, 10.84503681410932107018212785949
