| L(s) = 1 | + (−1.22 − 0.707i)2-s + (−0.328 − 2.98i)3-s + (0.999 + 1.73i)4-s + (1.93 + 1.11i)5-s + (−1.70 + 3.88i)6-s + (−5.45 + 4.39i)7-s − 2.82i·8-s + (−8.78 + 1.96i)9-s + (−1.58 − 2.73i)10-s + (−3.43 + 1.98i)11-s + (4.83 − 3.55i)12-s − 22.5·13-s + (9.78 − 1.52i)14-s + (2.69 − 6.14i)15-s + (−2.00 + 3.46i)16-s + (13.7 − 7.92i)17-s + ⋯ |
| L(s) = 1 | + (−0.612 − 0.353i)2-s + (−0.109 − 0.993i)3-s + (0.249 + 0.433i)4-s + (0.387 + 0.223i)5-s + (−0.284 + 0.647i)6-s + (−0.778 + 0.627i)7-s − 0.353i·8-s + (−0.975 + 0.217i)9-s + (−0.158 − 0.273i)10-s + (−0.312 + 0.180i)11-s + (0.402 − 0.295i)12-s − 1.73·13-s + (0.698 − 0.108i)14-s + (0.179 − 0.409i)15-s + (−0.125 + 0.216i)16-s + (0.807 − 0.465i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.321 - 0.946i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.321 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.112434 + 0.156962i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.112434 + 0.156962i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.22 + 0.707i)T \) |
| 3 | \( 1 + (0.328 + 2.98i)T \) |
| 5 | \( 1 + (-1.93 - 1.11i)T \) |
| 7 | \( 1 + (5.45 - 4.39i)T \) |
| good | 11 | \( 1 + (3.43 - 1.98i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + 22.5T + 169T^{2} \) |
| 17 | \( 1 + (-13.7 + 7.92i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (14.1 - 24.4i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (8.65 + 4.99i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 - 23.4iT - 841T^{2} \) |
| 31 | \( 1 + (-2.22 - 3.84i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-11.1 + 19.3i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 37.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 33.1T + 1.84e3T^{2} \) |
| 47 | \( 1 + (64.3 + 37.1i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-62.7 + 36.2i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (52.8 - 30.5i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-15.5 + 26.8i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-0.891 - 1.54i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 89.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (7.44 + 12.8i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (44.9 - 77.7i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 - 42.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (126. + 73.2i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + 162.T + 9.40e3T^{2} \) |
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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
−12.40431642234319793358690787366, −11.71189914648447474256309792674, −10.26967883611125589553830863094, −9.651008245740480025435483308771, −8.404713055487185108900251982121, −7.42708305900914744794462283636, −6.48883766131725634516636861794, −5.34000946200760226626179996911, −3.02901225039103289031754597738, −1.99470028462486813603680550023,
0.12220544819053103209222001352, 2.75366170979928668130675648735, 4.41913180886274653836486541834, 5.54324533464896295395375429793, 6.71090139977302449923604364401, 7.921315965735824986353161832741, 9.151641091113253747685680888164, 9.923359070643104425614886417905, 10.39330446836404599425396987507, 11.61680517645826032516348010724
