| 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
−14.61930536142073816526605869714, −13.51094866485141414757187710043, −12.23657956372858113918905303132, −11.00791349952214202866408750106, −10.15104581745285019956329408118, −9.056897213204604630687227936714, −7.38523158345082403810583065011, −6.12585123004639406477552507481, −4.59054099746462233744855284799, −2.75638589179082588374520108297,
1.20507070873966396133681926300, 3.88147411649146985791730754836, 5.60665582072197972506505004019, 6.93864483275479413510672877831, 8.185133905428785416315992368851, 9.538055671985091424609132916580, 10.64291295636422701792537749645, 12.28842342867425514963000993843, 12.73893491068893998714059058136, 13.79591186913841605172220928295
