L(s) = 1 | + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (−0.923 + 2.03i)5-s + (−2.56 + 0.634i)7-s + (−0.707 − 0.707i)8-s + (−2.20 − 0.365i)10-s + (4.28 − 2.47i)11-s + (−4.34 + 4.34i)13-s + (−1.27 − 2.31i)14-s + (0.500 − 0.866i)16-s + (−4.25 − 1.13i)17-s + (−3.22 − 1.86i)19-s + (−0.218 − 2.22i)20-s + (3.49 + 3.49i)22-s + (1.87 − 0.503i)23-s + ⋯ |
L(s) = 1 | + (0.183 + 0.683i)2-s + (−0.433 + 0.249i)4-s + (−0.413 + 0.910i)5-s + (−0.970 + 0.239i)7-s + (−0.249 − 0.249i)8-s + (−0.697 − 0.115i)10-s + (1.29 − 0.745i)11-s + (−1.20 + 1.20i)13-s + (−0.341 − 0.619i)14-s + (0.125 − 0.216i)16-s + (−1.03 − 0.276i)17-s + (−0.740 − 0.427i)19-s + (−0.0487 − 0.497i)20-s + (0.745 + 0.745i)22-s + (0.391 − 0.104i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.823 + 0.567i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.823 + 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.140789 - 0.452709i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.140789 - 0.452709i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.258 - 0.965i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.923 - 2.03i)T \) |
| 7 | \( 1 + (2.56 - 0.634i)T \) |
good | 11 | \( 1 + (-4.28 + 2.47i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (4.34 - 4.34i)T - 13iT^{2} \) |
| 17 | \( 1 + (4.25 + 1.13i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (3.22 + 1.86i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.87 + 0.503i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + 3.56T + 29T^{2} \) |
| 31 | \( 1 + (0.272 + 0.471i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.62 - 0.435i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + 0.268iT - 41T^{2} \) |
| 43 | \( 1 + (8.39 - 8.39i)T - 43iT^{2} \) |
| 47 | \( 1 + (-0.240 - 0.896i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-2.06 + 7.70i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-4.90 - 8.48i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.71 - 9.89i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.0109 - 0.0408i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 16.3iT - 71T^{2} \) |
| 73 | \( 1 + (-8.42 - 2.25i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (6.93 + 4.00i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.30 + 1.30i)T + 83iT^{2} \) |
| 89 | \( 1 + (0.229 - 0.397i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (10.6 + 10.6i)T + 97iT^{2} \) |
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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
−11.36224909359575203009301755429, −10.06277243573213648325473347851, −9.230002323326398533480536241010, −8.597472195678991717543784902884, −7.10351579897066979789852384119, −6.80079234751354076815129595326, −6.05557208304186331742927891147, −4.55189227942426714241837221838, −3.68484659989845170839709314804, −2.50778876913128184900021888455,
0.23240366932812762288430922944, 1.91795621378654407828021240208, 3.43556698932042370401431077327, 4.28556901177854073134295440117, 5.21102687000957040190624390592, 6.45480554345956208837427143480, 7.42773080696313430593529927853, 8.596377054767603402356508880656, 9.371726788221815699403590127657, 10.00305625558827094795368679856