L(s) = 1 | + (2.29 − 4.65i)3-s + (6.03 − 10.4i)5-s + (2.10 − 18.4i)7-s + (−16.4 − 21.4i)9-s + (0.00221 − 0.00127i)11-s + (6.06 + 3.50i)13-s + (−34.8 − 52.1i)15-s − 28.3·17-s + 49.1i·19-s + (−80.9 − 52.1i)21-s + (44.4 + 25.6i)23-s + (−10.3 − 17.9i)25-s + (−137. + 27.2i)27-s + (97.9 − 56.5i)29-s + (−28.4 − 16.4i)31-s + ⋯ |
L(s) = 1 | + (0.442 − 0.896i)3-s + (0.539 − 0.935i)5-s + (0.113 − 0.993i)7-s + (−0.608 − 0.793i)9-s + (6.06e−5 − 3.50e−5i)11-s + (0.129 + 0.0747i)13-s + (−0.599 − 0.897i)15-s − 0.403·17-s + 0.594i·19-s + (−0.840 − 0.541i)21-s + (0.403 + 0.232i)23-s + (−0.0828 − 0.143i)25-s + (−0.980 + 0.194i)27-s + (0.627 − 0.362i)29-s + (−0.164 − 0.0950i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.655i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.755 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.038879931\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.038879931\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.29 + 4.65i)T \) |
| 7 | \( 1 + (-2.10 + 18.4i)T \) |
good | 5 | \( 1 + (-6.03 + 10.4i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-0.00221 + 0.00127i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-6.06 - 3.50i)T + (1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + 28.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 49.1iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (-44.4 - 25.6i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-97.9 + 56.5i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (28.4 + 16.4i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 101.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (11.2 - 19.4i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (227. + 393. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-231. - 400. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + 567. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (145. - 252. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-592. + 341. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-269. + 467. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 307. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 495. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + (324. + 562. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-565. - 979. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 - 130.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-1.26e3 + 727. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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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
−11.37584750998377132204074031484, −10.16417739072800440344824184738, −9.126485186836653846485743591346, −8.295249892917544649257410454737, −7.30598593872263220771100219080, −6.29983599878737392957540317951, −5.02512139178674247586607585855, −3.62493799975510319636284518445, −1.89535124712364907029204909402, −0.76839196468662343543506851452,
2.30093964153191064397005126692, 3.16224116393601060815406608509, 4.71289736386552112197708186430, 5.79326386050900889732246411460, 6.90588087847905545782642590320, 8.373054909816617354576018534213, 9.110495777923903787231037550062, 10.08329329331966687284717369068, 10.85121108204236385356098113569, 11.71766808499092398938543140369