| L(s) = 1 | + (1.85 + 0.162i)2-s + (1.71 + 0.207i)3-s + (1.46 + 0.257i)4-s + (−2.23 − 0.123i)5-s + (3.16 + 0.665i)6-s + (−2.56 − 1.79i)7-s + (−0.928 − 0.248i)8-s + (2.91 + 0.712i)9-s + (−4.13 − 0.592i)10-s + (−1.21 + 3.34i)11-s + (2.46 + 0.746i)12-s + (0.165 + 1.89i)13-s + (−4.47 − 3.75i)14-s + (−3.81 − 0.675i)15-s + (−4.47 − 1.62i)16-s + (3.30 − 0.886i)17-s + ⋯ |
| L(s) = 1 | + (1.31 + 0.115i)2-s + (0.992 + 0.119i)3-s + (0.731 + 0.128i)4-s + (−0.998 − 0.0552i)5-s + (1.29 + 0.271i)6-s + (−0.969 − 0.678i)7-s + (−0.328 − 0.0879i)8-s + (0.971 + 0.237i)9-s + (−1.30 − 0.187i)10-s + (−0.367 + 1.00i)11-s + (0.710 + 0.215i)12-s + (0.0460 + 0.525i)13-s + (−1.19 − 1.00i)14-s + (−0.984 − 0.174i)15-s + (−1.11 − 0.407i)16-s + (0.802 − 0.215i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.136i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.136i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.06477 + 0.141331i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.06477 + 0.141331i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.71 - 0.207i)T \) |
| 5 | \( 1 + (2.23 + 0.123i)T \) |
| good | 2 | \( 1 + (-1.85 - 0.162i)T + (1.96 + 0.347i)T^{2} \) |
| 7 | \( 1 + (2.56 + 1.79i)T + (2.39 + 6.57i)T^{2} \) |
| 11 | \( 1 + (1.21 - 3.34i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.165 - 1.89i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (-3.30 + 0.886i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-5.00 + 2.89i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.749 - 1.06i)T + (-7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (-0.0144 + 0.0121i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.260 + 1.47i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (2.61 + 9.77i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (6.39 - 7.61i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (3.23 + 6.93i)T + (-27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (6.31 - 9.02i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (-3.54 + 3.54i)T - 53iT^{2} \) |
| 59 | \( 1 + (5.27 - 1.91i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.47 - 8.37i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-0.222 + 0.0194i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (0.428 + 0.247i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.07 - 7.75i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-1.13 - 1.35i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-0.643 + 7.35i)T + (-81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (-4.43 - 7.67i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.93 + 3.23i)T + (62.3 - 74.3i)T^{2} \) |
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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
−13.30719324193380991295020632071, −12.62983766095869790251342901835, −11.66081544712119554198126650324, −10.06419587222292275699767515501, −9.141231604257414573953693117433, −7.54587854536516458803299121678, −6.89211433400249391391585409144, −4.95247037308603826303734419289, −3.90731917050001536326702049695, −3.05749514687690403385815632979,
3.19863698954609558208480945523, 3.38793365266441258219290489419, 5.16228772771151321421339890440, 6.47091826696450082960124605515, 7.924696862454274532208887961966, 8.824093042800332112938484099473, 10.16595290732148714822660825282, 11.74285364394115304776351233517, 12.42738680724066448203028613554, 13.25492742956332023337814262103
