L(s) = 1 | + (0.221 − 1.39i)2-s + (0.687 − 0.350i)3-s + (−1.90 − 0.618i)4-s + (2.80 − 4.13i)5-s + (−0.337 − 1.03i)6-s + (2.38 − 2.38i)7-s + (−1.28 + 2.52i)8-s + (−4.94 + 6.79i)9-s + (−5.15 − 4.83i)10-s + (1.33 − 0.969i)11-s + (−1.52 + 0.241i)12-s + (2.37 + 14.9i)13-s + (−2.80 − 3.86i)14-s + (0.479 − 3.82i)15-s + (3.23 + 2.35i)16-s + (19.4 + 9.92i)17-s + ⋯ |
L(s) = 1 | + (0.110 − 0.698i)2-s + (0.229 − 0.116i)3-s + (−0.475 − 0.154i)4-s + (0.561 − 0.827i)5-s + (−0.0562 − 0.172i)6-s + (0.340 − 0.340i)7-s + (−0.160 + 0.315i)8-s + (−0.548 + 0.755i)9-s + (−0.515 − 0.483i)10-s + (0.121 − 0.0881i)11-s + (−0.127 + 0.0201i)12-s + (0.182 + 1.15i)13-s + (−0.200 − 0.275i)14-s + (0.0319 − 0.255i)15-s + (0.202 + 0.146i)16-s + (1.14 + 0.583i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.340 + 0.940i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.340 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.05110 - 0.737043i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.05110 - 0.737043i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.221 + 1.39i)T \) |
| 5 | \( 1 + (-2.80 + 4.13i)T \) |
good | 3 | \( 1 + (-0.687 + 0.350i)T + (5.29 - 7.28i)T^{2} \) |
| 7 | \( 1 + (-2.38 + 2.38i)T - 49iT^{2} \) |
| 11 | \( 1 + (-1.33 + 0.969i)T + (37.3 - 115. i)T^{2} \) |
| 13 | \( 1 + (-2.37 - 14.9i)T + (-160. + 52.2i)T^{2} \) |
| 17 | \( 1 + (-19.4 - 9.92i)T + (169. + 233. i)T^{2} \) |
| 19 | \( 1 + (24.0 - 7.79i)T + (292. - 212. i)T^{2} \) |
| 23 | \( 1 + (-0.856 - 0.135i)T + (503. + 163. i)T^{2} \) |
| 29 | \( 1 + (6.29 + 2.04i)T + (680. + 494. i)T^{2} \) |
| 31 | \( 1 + (16.2 + 49.9i)T + (-777. + 564. i)T^{2} \) |
| 37 | \( 1 + (-26.2 + 4.15i)T + (1.30e3 - 423. i)T^{2} \) |
| 41 | \( 1 + (-5.71 - 4.15i)T + (519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 + (56.4 + 56.4i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-15.4 - 30.3i)T + (-1.29e3 + 1.78e3i)T^{2} \) |
| 53 | \( 1 + (-24.8 + 12.6i)T + (1.65e3 - 2.27e3i)T^{2} \) |
| 59 | \( 1 + (-18.2 + 25.0i)T + (-1.07e3 - 3.31e3i)T^{2} \) |
| 61 | \( 1 + (82.7 - 60.1i)T + (1.14e3 - 3.53e3i)T^{2} \) |
| 67 | \( 1 + (33.4 + 17.0i)T + (2.63e3 + 3.63e3i)T^{2} \) |
| 71 | \( 1 + (29.9 - 92.2i)T + (-4.07e3 - 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-36.5 - 5.79i)T + (5.06e3 + 1.64e3i)T^{2} \) |
| 79 | \( 1 + (-96.9 - 31.4i)T + (5.04e3 + 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-69.2 + 135. i)T + (-4.04e3 - 5.57e3i)T^{2} \) |
| 89 | \( 1 + (63.1 + 86.8i)T + (-2.44e3 + 7.53e3i)T^{2} \) |
| 97 | \( 1 + (67.7 + 132. i)T + (-5.53e3 + 7.61e3i)T^{2} \) |
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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
−14.67288770718261550638801314414, −13.80076967801165701317697530809, −12.88210369562699059943496406306, −11.65130453460772151010014760834, −10.43423914955127089138272734734, −9.126415395083128037321145610934, −8.034223057282139663688456108312, −5.80787504288034578709308439293, −4.26231949884244189962986706542, −1.88272417976511088181627569198,
3.17455997252517298171897500097, 5.43260674906424544505036522276, 6.63019187418336558689201031189, 8.142147149185633956835702560989, 9.416750730789268275448920583976, 10.71051598338663302725143555053, 12.24182195414615118298909808634, 13.59476033981210810803708946045, 14.78883605700650939110915742059, 15.05604219012423128029610620524