L(s) = 1 | + (−0.349 + 0.605i)5-s + (0.229 − 0.132i)11-s + (1.13 + 0.657i)13-s − 3.72·17-s + 0.441i·19-s + (−4.29 − 2.48i)23-s + (2.25 + 3.90i)25-s + (0.273 − 0.157i)29-s + (−4.85 − 2.80i)31-s + 0.702·37-s + (5.39 − 9.34i)41-s + (3.73 + 6.46i)43-s + (−3.50 − 6.06i)47-s − 9.83i·53-s + 0.185i·55-s + ⋯ |
L(s) = 1 | + (−0.156 + 0.270i)5-s + (0.0692 − 0.0399i)11-s + (0.315 + 0.182i)13-s − 0.904·17-s + 0.101i·19-s + (−0.896 − 0.517i)23-s + (0.451 + 0.781i)25-s + (0.0507 − 0.0292i)29-s + (−0.872 − 0.503i)31-s + 0.115·37-s + (0.842 − 1.45i)41-s + (0.569 + 0.985i)43-s + (−0.510 − 0.884i)47-s − 1.35i·53-s + 0.0250i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.294 + 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.294 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8570659781\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8570659781\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 5 | \( 1 + (0.349 - 0.605i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.229 + 0.132i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.13 - 0.657i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 3.72T + 17T^{2} \) |
| 19 | \( 1 - 0.441iT - 19T^{2} \) |
| 23 | \( 1 + (4.29 + 2.48i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.273 + 0.157i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (4.85 + 2.80i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 0.702T + 37T^{2} \) |
| 41 | \( 1 + (-5.39 + 9.34i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.73 - 6.46i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (3.50 + 6.06i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 9.83iT - 53T^{2} \) |
| 59 | \( 1 + (6.73 - 11.6i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.89 - 2.82i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.97 + 5.14i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 13.4iT - 71T^{2} \) |
| 73 | \( 1 + 7.69iT - 73T^{2} \) |
| 79 | \( 1 + (0.698 + 1.20i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.72 - 6.45i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 11.1T + 89T^{2} \) |
| 97 | \( 1 + (9.18 - 5.30i)T + (48.5 - 84.0i)T^{2} \) |
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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
−7.893500308601105164948506812340, −7.30975693062598967191955149196, −6.51129837964575077371947131675, −5.94160293286285378833500828068, −5.03566198123390034501314731792, −4.18911120127577210863701577463, −3.54759231677134077010970095330, −2.52095945589610122102896736865, −1.67048777960493427815750758768, −0.23812833573828397108895137382,
1.10330467521976215848156246864, 2.17370950830499542087816745538, 3.10234401279720911036225616929, 4.07607352779368679631288943854, 4.62947828024429568207234890716, 5.55976542555005553238729682230, 6.25899852508193792989561213708, 6.94589111647104476627161246068, 7.81098924667462160557116861665, 8.328419823888493165082662650655