L(s) = 1 | + (0.385 − 0.222i)2-s + (−0.5 − 0.866i)3-s + (−0.900 + 1.56i)4-s − 0.246i·5-s + (−0.385 − 0.222i)6-s + (1.51 + 0.876i)7-s + 1.69i·8-s + (−0.499 + 0.866i)9-s + (−0.0549 − 0.0951i)10-s + (−4.89 + 2.82i)11-s + 1.80·12-s + 0.780·14-s + (−0.213 + 0.123i)15-s + (−1.42 − 2.46i)16-s + (−1.90 + 3.29i)17-s + 0.445i·18-s + ⋯ |
L(s) = 1 | + (0.272 − 0.157i)2-s + (−0.288 − 0.499i)3-s + (−0.450 + 0.780i)4-s − 0.110i·5-s + (−0.157 − 0.0908i)6-s + (0.573 + 0.331i)7-s + 0.598i·8-s + (−0.166 + 0.288i)9-s + (−0.0173 − 0.0301i)10-s + (−1.47 + 0.852i)11-s + 0.520·12-s + 0.208·14-s + (−0.0552 + 0.0318i)15-s + (−0.356 − 0.617i)16-s + (−0.461 + 0.798i)17-s + 0.104i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.114 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.114 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.614791 + 0.689985i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.614791 + 0.689985i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 + 0.866i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.385 + 0.222i)T + (1 - 1.73i)T^{2} \) |
| 5 | \( 1 + 0.246iT - 5T^{2} \) |
| 7 | \( 1 + (-1.51 - 0.876i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (4.89 - 2.82i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (1.90 - 3.29i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4.83 + 2.79i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.17 - 7.22i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.96 - 5.14i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 5.26iT - 31T^{2} \) |
| 37 | \( 1 + (2.76 - 1.59i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.385 + 0.222i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.856 + 1.48i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 6.73iT - 47T^{2} \) |
| 53 | \( 1 + 1.06T + 53T^{2} \) |
| 59 | \( 1 + (11.8 + 6.85i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.25 + 7.37i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.16 + 2.98i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.95 - 2.85i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 7.35iT - 73T^{2} \) |
| 79 | \( 1 - 4.45T + 79T^{2} \) |
| 83 | \( 1 + 10.1iT - 83T^{2} \) |
| 89 | \( 1 + (0.118 - 0.0685i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-11.8 - 6.84i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.15138373774175698715925143432, −10.59250197720214757031897280909, −9.141866934944685397948191189810, −8.370819162419067583300079149164, −7.62912959201150267339435052514, −6.69314405423215082351170247952, −5.11249379176899659657848214911, −4.80068316080582039943444404247, −3.16830665626593588014925429340, −1.98590194505190117031276684309,
0.51356859656640151804959839816, 2.63290388025957239082632648528, 4.25958159534674271864789747752, 4.93428156369790575453187745210, 5.82340390770154877502152166856, 6.77336010682308829244793312321, 8.133216584251195903834485900235, 8.876275667763187946156648086146, 10.04328282563969914532779597577, 10.71613045979604482107378119101