L(s) = 1 | − 4.72·7-s − 3.93i·11-s − 3.46i·13-s − 3.51·17-s + 5.44i·19-s − 7.11·23-s + 3.66i·29-s − 5.23·31-s − 0.414i·37-s − 3.00·41-s − 5.34i·43-s − 0.925·47-s + 15.3·49-s + 0.233i·53-s + 14.3i·59-s + ⋯ |
L(s) = 1 | − 1.78·7-s − 1.18i·11-s − 0.961i·13-s − 0.852·17-s + 1.24i·19-s − 1.48·23-s + 0.681i·29-s − 0.940·31-s − 0.0681i·37-s − 0.469·41-s − 0.815i·43-s − 0.135·47-s + 2.18·49-s + 0.0320i·53-s + 1.87i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.826 - 0.563i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.826 - 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6583118870\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6583118870\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 4.72T + 7T^{2} \) |
| 11 | \( 1 + 3.93iT - 11T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 + 3.51T + 17T^{2} \) |
| 19 | \( 1 - 5.44iT - 19T^{2} \) |
| 23 | \( 1 + 7.11T + 23T^{2} \) |
| 29 | \( 1 - 3.66iT - 29T^{2} \) |
| 31 | \( 1 + 5.23T + 31T^{2} \) |
| 37 | \( 1 + 0.414iT - 37T^{2} \) |
| 41 | \( 1 + 3.00T + 41T^{2} \) |
| 43 | \( 1 + 5.34iT - 43T^{2} \) |
| 47 | \( 1 + 0.925T + 47T^{2} \) |
| 53 | \( 1 - 0.233iT - 53T^{2} \) |
| 59 | \( 1 - 14.3iT - 59T^{2} \) |
| 61 | \( 1 + 0.118iT - 61T^{2} \) |
| 67 | \( 1 + 13.4iT - 67T^{2} \) |
| 71 | \( 1 - 2.19T + 71T^{2} \) |
| 73 | \( 1 + 0.563T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 - 11.3iT - 83T^{2} \) |
| 89 | \( 1 + 8.88T + 89T^{2} \) |
| 97 | \( 1 - 7.27T + 97T^{2} \) |
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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
−8.084223047086459517401399590459, −7.20329996207698568170203804017, −6.50760844507763416635187904933, −5.82928047902470112228593703783, −5.56199614031329712731452795384, −4.16634039622184700019424307265, −3.48776179915069013209495938981, −3.05506051421334354308901473408, −1.96732009032546972029252267859, −0.54141323676718925279763665938,
0.27629138960160382909090392576, 1.93490272302299396812333381828, 2.53078845195904581919247439215, 3.56077561177411340309357924827, 4.21453798452504862754506439849, 4.90363573353218518342189684337, 5.98877947390465532654973783049, 6.59733352479742238692119085596, 6.94156745401602243437115029639, 7.70078049767479835098964493152