| L(s) = 1 | + (0.306 + 1.70i)3-s + 2.23i·5-s + 2.64·7-s + (−2.81 + 1.04i)9-s + 5.55i·11-s − 7.13·13-s + (−3.81 + 0.686i)15-s − 5.75i·17-s + (0.811 + 4.51i)21-s − 5.00·25-s + (−2.64 − 4.47i)27-s + 4.83i·29-s + (−9.47 + 1.70i)33-s + 5.91i·35-s + (−2.18 − 12.1i)39-s + ⋯ |
| L(s) = 1 | + (0.177 + 0.984i)3-s + 0.999i·5-s + 0.999·7-s + (−0.937 + 0.348i)9-s + 1.67i·11-s − 1.97·13-s + (−0.984 + 0.177i)15-s − 1.39i·17-s + (0.177 + 0.984i)21-s − 1.00·25-s + (−0.509 − 0.860i)27-s + 0.898i·29-s + (−1.64 + 0.296i)33-s + 0.999i·35-s + (−0.350 − 1.94i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.177i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 + 0.177i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.194522136\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.194522136\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.306 - 1.70i)T \) |
| 5 | \( 1 - 2.23iT \) |
| 7 | \( 1 - 2.64T \) |
| good | 11 | \( 1 - 5.55iT - 11T^{2} \) |
| 13 | \( 1 + 7.13T + 13T^{2} \) |
| 17 | \( 1 + 5.75iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 4.83iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 1.28iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 11.8iT - 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 - 14.8T + 79T^{2} \) |
| 83 | \( 1 - 8.94iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 3.45T + 97T^{2} \) |
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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
−9.699106458222289068735139330683, −9.381196446991947358117588169124, −8.065627991146959416736057710502, −7.34410137547533235250300416595, −6.87829664896970751996484965698, −5.23760738512818977292805530266, −4.93157951036119054249133063710, −4.05364627503159693998947500258, −2.72502920680113044411433687757, −2.18472959499529645016010715699,
0.41898726146491076680131555004, 1.58828859841099929987369309904, 2.54481033531360542667717350450, 3.86750407790646016181016830857, 4.97821292762474697443090738126, 5.65041799292954383898282761262, 6.47040662309942752855425836466, 7.75039075221181891359993985070, 7.996393159919406152295235061574, 8.715293866222593212817107402333
