L(s) = 1 | + (−0.573 + 0.819i)2-s + (0.190 − 2.17i)3-s + (−0.342 − 0.939i)4-s + (1.67 + 1.40i)6-s + (0.115 + 0.431i)7-s + (0.965 + 0.258i)8-s + (−1.74 − 0.307i)9-s + (2.37 + 4.12i)11-s + (−2.11 + 0.565i)12-s + (2.40 − 0.210i)13-s + (−0.419 − 0.152i)14-s + (−0.766 + 0.642i)16-s + (0.560 + 0.392i)17-s + (1.25 − 1.25i)18-s + (2.15 + 3.79i)19-s + ⋯ |
L(s) = 1 | + (−0.405 + 0.579i)2-s + (0.109 − 1.25i)3-s + (−0.171 − 0.469i)4-s + (0.683 + 0.573i)6-s + (0.0436 + 0.163i)7-s + (0.341 + 0.0915i)8-s + (−0.582 − 0.102i)9-s + (0.717 + 1.24i)11-s + (−0.609 + 0.163i)12-s + (0.666 − 0.0582i)13-s + (−0.112 − 0.0408i)14-s + (−0.191 + 0.160i)16-s + (0.135 + 0.0951i)17-s + (0.295 − 0.295i)18-s + (0.493 + 0.869i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.000757i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.000757i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.44285 - 0.000546452i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.44285 - 0.000546452i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.573 - 0.819i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-2.15 - 3.79i)T \) |
good | 3 | \( 1 + (-0.190 + 2.17i)T + (-2.95 - 0.520i)T^{2} \) |
| 7 | \( 1 + (-0.115 - 0.431i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-2.37 - 4.12i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.40 + 0.210i)T + (12.8 - 2.25i)T^{2} \) |
| 17 | \( 1 + (-0.560 - 0.392i)T + (5.81 + 15.9i)T^{2} \) |
| 23 | \( 1 + (-0.404 - 0.188i)T + (14.7 + 17.6i)T^{2} \) |
| 29 | \( 1 + (0.892 - 5.06i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-2.27 - 1.31i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.48 - 1.48i)T + 37iT^{2} \) |
| 41 | \( 1 + (0.333 + 0.397i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (3.94 + 8.46i)T + (-27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (-3.02 - 4.32i)T + (-16.0 + 44.1i)T^{2} \) |
| 53 | \( 1 + (-3.42 + 7.35i)T + (-34.0 - 40.6i)T^{2} \) |
| 59 | \( 1 + (-0.0555 - 0.315i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (7.77 - 2.82i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-9.20 + 6.44i)T + (22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (-0.879 + 2.41i)T + (-54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (4.32 + 0.378i)T + (71.8 + 12.6i)T^{2} \) |
| 79 | \( 1 + (-12.3 + 10.3i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-0.838 + 0.224i)T + (71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (-12.7 - 10.7i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (1.72 - 2.46i)T + (-33.1 - 91.1i)T^{2} \) |
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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
−9.877377131928123865277191233396, −9.006136117025558230966982516191, −8.190082383909103741073197298930, −7.42633227333299720373407828148, −6.81627029161985321680147172331, −6.08971136220864985821486025395, −5.01133412105830122355940217683, −3.71568344022853675781825822477, −2.05679380265802431943100130423, −1.19152004882849052071086395376,
0.991003718242224119253397255633, 2.82214315278888936700084110360, 3.71063540538416987282971821578, 4.39574984667344749333719423523, 5.54138584916875380883472132139, 6.59092160763960888825832098990, 7.84013385959105800670891170575, 8.760258566252243968446161621794, 9.282697865952749407031779003842, 9.991639026030380920797005197892