| L(s) = 1 | − 1.08i·3-s − 2.61·5-s + (−2.61 − 0.414i)7-s + 1.82·9-s + 2·11-s + 4.77·13-s + 2.82i·15-s − 3.06i·17-s − 4.14i·19-s + (−0.448 + 2.82i)21-s + 7.65i·23-s + 1.82·25-s − 5.22i·27-s − 3.65i·29-s − 3.06·31-s + ⋯ |
| L(s) = 1 | − 0.624i·3-s − 1.16·5-s + (−0.987 − 0.156i)7-s + 0.609·9-s + 0.603·11-s + 1.32·13-s + 0.730i·15-s − 0.742i·17-s − 0.950i·19-s + (−0.0978 + 0.617i)21-s + 1.59i·23-s + 0.365·25-s − 1.00i·27-s − 0.679i·29-s − 0.549·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.809 + 0.587i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.809 + 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8583111120\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8583111120\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (2.61 + 0.414i)T \) |
| good | 3 | \( 1 + 1.08iT - 3T^{2} \) |
| 5 | \( 1 + 2.61T + 5T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 4.77T + 13T^{2} \) |
| 17 | \( 1 + 3.06iT - 17T^{2} \) |
| 19 | \( 1 + 4.14iT - 19T^{2} \) |
| 23 | \( 1 - 7.65iT - 23T^{2} \) |
| 29 | \( 1 + 3.65iT - 29T^{2} \) |
| 31 | \( 1 + 3.06T + 31T^{2} \) |
| 37 | \( 1 + 7.65iT - 37T^{2} \) |
| 41 | \( 1 - 9.55iT - 41T^{2} \) |
| 43 | \( 1 + 3.65T + 43T^{2} \) |
| 47 | \( 1 + 7.39T + 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + 8.47iT - 59T^{2} \) |
| 61 | \( 1 + 2.61T + 61T^{2} \) |
| 67 | \( 1 + 15.6T + 67T^{2} \) |
| 71 | \( 1 + 8.82iT - 71T^{2} \) |
| 73 | \( 1 + 12.6iT - 73T^{2} \) |
| 79 | \( 1 + 12.8iT - 79T^{2} \) |
| 83 | \( 1 + 11.5iT - 83T^{2} \) |
| 89 | \( 1 + 2.16iT - 89T^{2} \) |
| 97 | \( 1 - 13.5iT - 97T^{2} \) |
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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
−9.074154041945882202578607125921, −7.901169778595977336983141917552, −7.44326844244771526493379392759, −6.66374548462052505372148309683, −6.04111671141550108960554105890, −4.67440586885167596362250886516, −3.76595324397513378230019649764, −3.19868257939613069899895185580, −1.57743717931795837972402843320, −0.35483099364878232545844217443,
1.36775382181584338458591232190, 3.13530677215445032150611345753, 3.92948778803145440600643660638, 4.20843925895011334568508514257, 5.59600230197385970541050044960, 6.50896713448273575739029656942, 7.09598703502070698583177227723, 8.297668750561796144902622084559, 8.669080848195873868851778141043, 9.634415835775727300507805458760
