| L(s) = 1 | + (−0.617 − 1.69i)3-s + (0.757 − 4.29i)5-s + (−2.41 − 4.17i)7-s + (4.39 − 3.68i)9-s + (0.0945 − 0.163i)11-s + (1.39 − 3.82i)13-s + (−7.75 + 1.36i)15-s + (13.1 + 11.0i)17-s + (−8.88 + 16.7i)19-s + (−5.59 + 6.66i)21-s + (2.45 + 13.9i)23-s + (5.63 + 2.04i)25-s + (−23.0 − 13.3i)27-s + (18.3 + 21.8i)29-s + (−5.92 + 3.42i)31-s + ⋯ |
| L(s) = 1 | + (−0.205 − 0.565i)3-s + (0.151 − 0.858i)5-s + (−0.344 − 0.596i)7-s + (0.488 − 0.409i)9-s + (0.00859 − 0.0148i)11-s + (0.107 − 0.294i)13-s + (−0.516 + 0.0911i)15-s + (0.771 + 0.647i)17-s + (−0.467 + 0.883i)19-s + (−0.266 + 0.317i)21-s + (0.106 + 0.604i)23-s + (0.225 + 0.0819i)25-s + (−0.853 − 0.492i)27-s + (0.633 + 0.754i)29-s + (−0.191 + 0.110i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.210 + 0.977i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.210 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.915055 - 0.739041i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.915055 - 0.739041i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (8.88 - 16.7i)T \) |
| good | 3 | \( 1 + (0.617 + 1.69i)T + (-6.89 + 5.78i)T^{2} \) |
| 5 | \( 1 + (-0.757 + 4.29i)T + (-23.4 - 8.55i)T^{2} \) |
| 7 | \( 1 + (2.41 + 4.17i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-0.0945 + 0.163i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-1.39 + 3.82i)T + (-129. - 108. i)T^{2} \) |
| 17 | \( 1 + (-13.1 - 11.0i)T + (50.1 + 284. i)T^{2} \) |
| 23 | \( 1 + (-2.45 - 13.9i)T + (-497. + 180. i)T^{2} \) |
| 29 | \( 1 + (-18.3 - 21.8i)T + (-146. + 828. i)T^{2} \) |
| 31 | \( 1 + (5.92 - 3.42i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 31.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-19.4 - 53.3i)T + (-1.28e3 + 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-7.16 + 40.6i)T + (-1.73e3 - 632. i)T^{2} \) |
| 47 | \( 1 + (-44.9 + 37.6i)T + (383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 + (5.32 - 0.938i)T + (2.63e3 - 960. i)T^{2} \) |
| 59 | \( 1 + (36.3 - 43.2i)T + (-604. - 3.42e3i)T^{2} \) |
| 61 | \( 1 + (15.7 + 89.1i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (-5.04 - 6.01i)T + (-779. + 4.42e3i)T^{2} \) |
| 71 | \( 1 + (-23.9 - 4.22i)T + (4.73e3 + 1.72e3i)T^{2} \) |
| 73 | \( 1 + (111. - 40.7i)T + (4.08e3 - 3.42e3i)T^{2} \) |
| 79 | \( 1 + (-47.8 - 131. i)T + (-4.78e3 + 4.01e3i)T^{2} \) |
| 83 | \( 1 + (-52.0 - 90.1i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + (52.7 - 144. i)T + (-6.06e3 - 5.09e3i)T^{2} \) |
| 97 | \( 1 + (52.4 - 62.4i)T + (-1.63e3 - 9.26e3i)T^{2} \) |
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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
−13.79591186913841605172220928295, −12.73893491068893998714059058136, −12.28842342867425514963000993843, −10.64291295636422701792537749645, −9.538055671985091424609132916580, −8.185133905428785416315992368851, −6.93864483275479413510672877831, −5.60665582072197972506505004019, −3.88147411649146985791730754836, −1.20507070873966396133681926300,
2.75638589179082588374520108297, 4.59054099746462233744855284799, 6.12585123004639406477552507481, 7.38523158345082403810583065011, 9.056897213204604630687227936714, 10.15104581745285019956329408118, 11.00791349952214202866408750106, 12.23657956372858113918905303132, 13.51094866485141414757187710043, 14.61930536142073816526605869714
