L(s) = 1 | + (0.456 − 0.166i)3-s + (0.485 − 2.75i)5-s + (1.68 + 2.91i)7-s + (−2.11 + 1.77i)9-s + (0.258 − 0.447i)11-s + (−4.37 − 1.59i)13-s + (−0.235 − 1.33i)15-s + (−0.735 − 0.617i)17-s + (−3.12 + 3.04i)19-s + (1.25 + 1.05i)21-s + (0.629 + 3.57i)23-s + (−2.63 − 0.958i)25-s + (−1.40 + 2.42i)27-s + (6.21 − 5.21i)29-s + (−2.38 − 4.13i)31-s + ⋯ |
L(s) = 1 | + (0.263 − 0.0960i)3-s + (0.216 − 1.23i)5-s + (0.636 + 1.10i)7-s + (−0.705 + 0.592i)9-s + (0.0778 − 0.134i)11-s + (−1.21 − 0.441i)13-s + (−0.0609 − 0.345i)15-s + (−0.178 − 0.149i)17-s + (−0.715 + 0.698i)19-s + (0.273 + 0.229i)21-s + (0.131 + 0.744i)23-s + (−0.526 − 0.191i)25-s + (−0.269 + 0.467i)27-s + (1.15 − 0.968i)29-s + (−0.429 − 0.743i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 + 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.00324 - 0.130669i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.00324 - 0.130669i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (3.12 - 3.04i)T \) |
good | 3 | \( 1 + (-0.456 + 0.166i)T + (2.29 - 1.92i)T^{2} \) |
| 5 | \( 1 + (-0.485 + 2.75i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-1.68 - 2.91i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.258 + 0.447i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (4.37 + 1.59i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (0.735 + 0.617i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (-0.629 - 3.57i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-6.21 + 5.21i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (2.38 + 4.13i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 9.13T + 37T^{2} \) |
| 41 | \( 1 + (6.54 - 2.38i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (0.817 - 4.63i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-10.4 + 8.74i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (1.20 + 6.81i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (10.9 + 9.19i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.05 - 6.00i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-3.38 + 2.84i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (1.66 - 9.42i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-5.65 + 2.05i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (9.85 - 3.58i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-2.39 - 4.14i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (8.30 + 3.02i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-8.46 - 7.10i)T + (16.8 + 95.5i)T^{2} \) |
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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
−14.51893043166139748598860062107, −13.31380960018966064760246189585, −12.33085278406011538286051206712, −11.44640651642312856118774761003, −9.760269343573532536189195768520, −8.641974788424511633014989962291, −7.940815014043327935710845037687, −5.73500631592716291786890309492, −4.82408695978744130000053124508, −2.30298806306697667701588489468,
2.77055465366898505700181350683, 4.48779107652015503272330983523, 6.51722131074141707771854491724, 7.40473329084772466965499844888, 8.936519968234725533826720167201, 10.34028495820404225123472025460, 11.02212408631200887187996097394, 12.32317581276490526579771084095, 13.98299111261948096762027070493, 14.40976802076476336003589989946