L(s) = 1 | + 3.63i·5-s + (1 + 2.44i)7-s + 1.50i·11-s + 5.91i·13-s − 5.14i·17-s − 5.24·19-s + 8.78i·23-s − 8.24·25-s + 8.91·29-s − 3.24·31-s + (−8.91 + 3.63i)35-s + 37-s + 6.65i·41-s − 9.37i·43-s − 3.69·47-s + ⋯ |
L(s) = 1 | + 1.62i·5-s + (0.377 + 0.925i)7-s + 0.454i·11-s + 1.64i·13-s − 1.24i·17-s − 1.20·19-s + 1.83i·23-s − 1.64·25-s + 1.65·29-s − 0.582·31-s + (−1.50 + 0.615i)35-s + 0.164·37-s + 1.03i·41-s − 1.43i·43-s − 0.538·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.135i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.513396310\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.513396310\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1 - 2.44i)T \) |
good | 5 | \( 1 - 3.63iT - 5T^{2} \) |
| 11 | \( 1 - 1.50iT - 11T^{2} \) |
| 13 | \( 1 - 5.91iT - 13T^{2} \) |
| 17 | \( 1 + 5.14iT - 17T^{2} \) |
| 19 | \( 1 + 5.24T + 19T^{2} \) |
| 23 | \( 1 - 8.78iT - 23T^{2} \) |
| 29 | \( 1 - 8.91T + 29T^{2} \) |
| 31 | \( 1 + 3.24T + 31T^{2} \) |
| 37 | \( 1 - T + 37T^{2} \) |
| 41 | \( 1 - 6.65iT - 41T^{2} \) |
| 43 | \( 1 + 9.37iT - 43T^{2} \) |
| 47 | \( 1 + 3.69T + 47T^{2} \) |
| 53 | \( 1 - 12.6T + 53T^{2} \) |
| 59 | \( 1 - 8.91T + 59T^{2} \) |
| 61 | \( 1 + 3.46iT - 61T^{2} \) |
| 67 | \( 1 + 10.8iT - 67T^{2} \) |
| 71 | \( 1 + 3.63iT - 71T^{2} \) |
| 73 | \( 1 - 0.420iT - 73T^{2} \) |
| 79 | \( 1 + 4.47iT - 79T^{2} \) |
| 83 | \( 1 - 3.69T + 83T^{2} \) |
| 89 | \( 1 + 13.9iT - 89T^{2} \) |
| 97 | \( 1 - 13.2iT - 97T^{2} \) |
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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
−9.202306100618040090175276249920, −8.330250316361748491776675050532, −7.35265042036705368821089106352, −6.84727111942860485937026068181, −6.26625124856155909717617274561, −5.28144398480595824472595878238, −4.38742206716570776523408786395, −3.40115118862095487011183889496, −2.46271910114166375219419299602, −1.87774330177259917140852766999,
0.50357090938047637364049492536, 1.19749245138678882116307096611, 2.53056085332672203852734687000, 3.89460281954542998628946986677, 4.40807325693662215342607286192, 5.21170423519660237421085735609, 5.95661307145539828861490768358, 6.82856033874754099043127073003, 8.018144151929269794329277376538, 8.412025291429080600902299996596