| L(s) = 1 | + (−2.19 − 0.466i)2-s + (0.199 − 1.89i)3-s + (2.77 + 1.23i)4-s + (1.64 + 1.83i)5-s + (−1.32 + 4.07i)6-s + (0.551 − 2.58i)7-s + (−1.89 − 1.37i)8-s + (−0.630 − 0.134i)9-s + (−2.76 − 4.79i)10-s + (2.90 − 5.02i)12-s + (0.0807 + 0.248i)13-s + (−2.41 + 5.42i)14-s + (3.80 − 2.76i)15-s + (−0.556 − 0.617i)16-s + (−6.69 + 1.42i)17-s + (1.32 + 0.588i)18-s + ⋯ |
| L(s) = 1 | + (−1.55 − 0.330i)2-s + (0.115 − 1.09i)3-s + (1.38 + 0.618i)4-s + (0.737 + 0.819i)5-s + (−0.540 + 1.66i)6-s + (0.208 − 0.978i)7-s + (−0.669 − 0.486i)8-s + (−0.210 − 0.0446i)9-s + (−0.875 − 1.51i)10-s + (0.838 − 1.45i)12-s + (0.0223 + 0.0689i)13-s + (−0.646 + 1.45i)14-s + (0.983 − 0.714i)15-s + (−0.139 − 0.154i)16-s + (−1.62 + 0.344i)17-s + (0.311 + 0.138i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.401 + 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.401 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.456761 - 0.698872i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.456761 - 0.698872i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-0.551 + 2.58i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + (2.19 + 0.466i)T + (1.82 + 0.813i)T^{2} \) |
| 3 | \( 1 + (-0.199 + 1.89i)T + (-2.93 - 0.623i)T^{2} \) |
| 5 | \( 1 + (-1.64 - 1.83i)T + (-0.522 + 4.97i)T^{2} \) |
| 13 | \( 1 + (-0.0807 - 0.248i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (6.69 - 1.42i)T + (15.5 - 6.91i)T^{2} \) |
| 19 | \( 1 + (-3.01 + 1.34i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (0.119 - 0.207i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-6.23 + 4.53i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (0.527 - 0.586i)T + (-3.24 - 30.8i)T^{2} \) |
| 37 | \( 1 + (0.600 + 5.71i)T + (-36.1 + 7.69i)T^{2} \) |
| 41 | \( 1 + (-0.572 - 0.415i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 9.16T + 43T^{2} \) |
| 47 | \( 1 + (-5.44 + 2.42i)T + (31.4 - 34.9i)T^{2} \) |
| 53 | \( 1 + (-4.29 + 4.77i)T + (-5.54 - 52.7i)T^{2} \) |
| 59 | \( 1 + (-1.62 - 0.721i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + (-2.58 - 2.86i)T + (-6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (5.41 + 9.38i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (3.76 - 11.5i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (12.6 + 5.61i)T + (48.8 + 54.2i)T^{2} \) |
| 79 | \( 1 + (-3.82 - 0.814i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (-3.04 + 9.38i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (4.13 - 7.16i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.516 - 1.58i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.02552130801935482090404701615, −9.075921630574405706833938485761, −8.231815520950755743024447756093, −7.31902008743419572574358651419, −6.96866021276730363415777866831, −6.14112483357845986202749461862, −4.36787707726898743087136814658, −2.62879319370572559563267121810, −1.91700400403263353278453158848, −0.73489485890875557794752735687,
1.34826504354165654559553018259, 2.61662093819036385159646278534, 4.37395787986491432801075640320, 5.20974698511540258902652189086, 6.19466443452923938657150255344, 7.25893112434375501806945329818, 8.445118121942344702334435740493, 9.037277102814954207827684322337, 9.287111943795702785384604818593, 10.14181091856743596832772747683
