| L(s) = 1 | − 0.304i·2-s − i·3-s + 1.90·4-s + (0.355 − 2.20i)5-s − 0.304·6-s − i·7-s − 1.18i·8-s − 9-s + (−0.671 − 0.108i)10-s − 6.38·11-s − 1.90i·12-s + 1.12i·13-s − 0.304·14-s + (−2.20 − 0.355i)15-s + 3.45·16-s − i·17-s + ⋯ |
| L(s) = 1 | − 0.215i·2-s − 0.577i·3-s + 0.953·4-s + (0.158 − 0.987i)5-s − 0.124·6-s − 0.377i·7-s − 0.420i·8-s − 0.333·9-s + (−0.212 − 0.0341i)10-s − 1.92·11-s − 0.550i·12-s + 0.312i·13-s − 0.0813·14-s + (−0.570 − 0.0916i)15-s + 0.863·16-s − 0.242i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1785 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.158i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1785 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 - 0.158i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.167148325\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.167148325\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + iT \) |
| 5 | \( 1 + (-0.355 + 2.20i)T \) |
| 7 | \( 1 + iT \) |
| 17 | \( 1 + iT \) |
| good | 2 | \( 1 + 0.304iT - 2T^{2} \) |
| 11 | \( 1 + 6.38T + 11T^{2} \) |
| 13 | \( 1 - 1.12iT - 13T^{2} \) |
| 19 | \( 1 + 5.66T + 19T^{2} \) |
| 23 | \( 1 - 0.581iT - 23T^{2} \) |
| 29 | \( 1 + 3.76T + 29T^{2} \) |
| 31 | \( 1 - 0.646T + 31T^{2} \) |
| 37 | \( 1 - 5.95iT - 37T^{2} \) |
| 41 | \( 1 + 5.31T + 41T^{2} \) |
| 43 | \( 1 + 4.23iT - 43T^{2} \) |
| 47 | \( 1 + 5.25iT - 47T^{2} \) |
| 53 | \( 1 + 13.3iT - 53T^{2} \) |
| 59 | \( 1 - 6.85T + 59T^{2} \) |
| 61 | \( 1 - 14.0T + 61T^{2} \) |
| 67 | \( 1 - 6.03iT - 67T^{2} \) |
| 71 | \( 1 + 8.66T + 71T^{2} \) |
| 73 | \( 1 + 9.19iT - 73T^{2} \) |
| 79 | \( 1 - 4.17T + 79T^{2} \) |
| 83 | \( 1 - 10.0iT - 83T^{2} \) |
| 89 | \( 1 - 15.5T + 89T^{2} \) |
| 97 | \( 1 + 12.5iT - 97T^{2} \) |
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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
−8.578860508093042786805374739607, −8.123382729593521244407586916672, −7.30023119287730498547507069564, −6.57681972569510620767441192572, −5.59339763725375980496336983943, −4.94329710165077794367201898805, −3.68446236209746252351084467474, −2.46088929123106853936319877471, −1.80018175227065106329118150079, −0.36939585687380614947259679685,
2.22409793754747032703751941227, 2.66879751712040475394040009070, 3.66848922362252952440260508693, 5.01851727651139635196186940347, 5.79923468061988693514387421919, 6.37901343247972712409562429276, 7.43854732138004211014357983294, 7.918360997139930132502077524765, 8.825025481993050207529217948326, 10.04259530047981185235677149328
