| L(s) = 1 | + i·2-s + (−1.73 + 0.0828i)3-s − 4-s + (−0.0828 − 1.73i)6-s − 4.34·7-s − i·8-s + (2.98 − 0.286i)9-s + (−1.20 − 3.09i)11-s + (1.73 − 0.0828i)12-s + 4.51·13-s − 4.34i·14-s + 16-s + 2.72i·17-s + (0.286 + 2.98i)18-s − 3.97i·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + (−0.998 + 0.0478i)3-s − 0.5·4-s + (−0.0338 − 0.706i)6-s − 1.64·7-s − 0.353i·8-s + (0.995 − 0.0956i)9-s + (−0.363 − 0.931i)11-s + (0.499 − 0.0239i)12-s + 1.25·13-s − 1.16i·14-s + 0.250·16-s + 0.659i·17-s + (0.0676 + 0.703i)18-s − 0.911i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.772 - 0.634i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.772 - 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4927807942\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4927807942\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (1.73 - 0.0828i)T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + (1.20 + 3.09i)T \) |
| good | 7 | \( 1 + 4.34T + 7T^{2} \) |
| 13 | \( 1 - 4.51T + 13T^{2} \) |
| 17 | \( 1 - 2.72iT - 17T^{2} \) |
| 19 | \( 1 + 3.97iT - 19T^{2} \) |
| 23 | \( 1 - 1.66T + 23T^{2} \) |
| 29 | \( 1 + 5.08T + 29T^{2} \) |
| 31 | \( 1 - 9.74T + 31T^{2} \) |
| 37 | \( 1 - 5.60iT - 37T^{2} \) |
| 41 | \( 1 + 2.92T + 41T^{2} \) |
| 43 | \( 1 + 3.25T + 43T^{2} \) |
| 47 | \( 1 + 9.25T + 47T^{2} \) |
| 53 | \( 1 + 3.05T + 53T^{2} \) |
| 59 | \( 1 - 5.43iT - 59T^{2} \) |
| 61 | \( 1 - 11.5iT - 61T^{2} \) |
| 67 | \( 1 + 12.3iT - 67T^{2} \) |
| 71 | \( 1 - 3.60iT - 71T^{2} \) |
| 73 | \( 1 + 1.36T + 73T^{2} \) |
| 79 | \( 1 - 1.07iT - 79T^{2} \) |
| 83 | \( 1 - 14.5iT - 83T^{2} \) |
| 89 | \( 1 + 3.75iT - 89T^{2} \) |
| 97 | \( 1 - 12.7iT - 97T^{2} \) |
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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
−9.697892898293679402647779463697, −8.863304968500249200499693200160, −8.077152104933055202114491593586, −6.89680483927801344966159944794, −6.35590173078605925217812017800, −5.94836983238186773663523726721, −4.98277729794460876762127390047, −3.86575580874103757297644814147, −3.07936742138287968456641472600, −0.963922454946368981265279200197,
0.28245256474278655269830147612, 1.68309073028164611961435363809, 3.07932777942260336305572560686, 3.89747461208879109168258011985, 4.87902338286238352711062973190, 5.86542322732774963645063702573, 6.49470658668970036419652319868, 7.28793556273717314858716800032, 8.384993382986769199166957806193, 9.586551795089795615804897814923
