| L(s) = 1 | + (0.598 + 2.23i)2-s + (3.51 + 2.03i)3-s + (−1.17 + 0.676i)4-s + (−2.43 + 9.07i)6-s + (−2.56 + 9.58i)7-s + (4.32 + 4.32i)8-s + (3.74 + 6.49i)9-s + (−2.41 − 9.02i)11-s − 5.49·12-s + (12.8 + 1.86i)13-s − 22.9·14-s + (−9.79 + 16.9i)16-s + (−7.95 − 13.7i)17-s + (−12.2 + 12.2i)18-s + (6.92 − 25.8i)19-s + ⋯ |
| L(s) = 1 | + (0.299 + 1.11i)2-s + (1.17 + 0.676i)3-s + (−0.293 + 0.169i)4-s + (−0.405 + 1.51i)6-s + (−0.366 + 1.36i)7-s + (0.541 + 0.541i)8-s + (0.416 + 0.721i)9-s + (−0.219 − 0.820i)11-s − 0.458·12-s + (0.989 + 0.143i)13-s − 1.63·14-s + (−0.611 + 1.05i)16-s + (−0.467 − 0.810i)17-s + (−0.681 + 0.681i)18-s + (0.364 − 1.35i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.763 - 0.645i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.763 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.01229 + 2.76570i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01229 + 2.76570i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 + (-12.8 - 1.86i)T \) |
| good | 2 | \( 1 + (-0.598 - 2.23i)T + (-3.46 + 2i)T^{2} \) |
| 3 | \( 1 + (-3.51 - 2.03i)T + (4.5 + 7.79i)T^{2} \) |
| 7 | \( 1 + (2.56 - 9.58i)T + (-42.4 - 24.5i)T^{2} \) |
| 11 | \( 1 + (2.41 + 9.02i)T + (-104. + 60.5i)T^{2} \) |
| 17 | \( 1 + (7.95 + 13.7i)T + (-144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-6.92 + 25.8i)T + (-312. - 180.5i)T^{2} \) |
| 23 | \( 1 + (7.99 - 13.8i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (16.4 - 28.4i)T + (-420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (-11.4 - 11.4i)T + 961iT^{2} \) |
| 37 | \( 1 + (31.1 - 8.33i)T + (1.18e3 - 684.5i)T^{2} \) |
| 41 | \( 1 + (-66.9 + 17.9i)T + (1.45e3 - 840.5i)T^{2} \) |
| 43 | \( 1 + (7.35 + 12.7i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-16.7 - 16.7i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + 87.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (14.9 + 4.01i)T + (3.01e3 + 1.74e3i)T^{2} \) |
| 61 | \( 1 + (14.4 + 25.0i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-2.34 - 8.75i)T + (-3.88e3 + 2.24e3i)T^{2} \) |
| 71 | \( 1 + (-1.21 + 4.51i)T + (-4.36e3 - 2.52e3i)T^{2} \) |
| 73 | \( 1 + (-95.0 - 95.0i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 + 14.6T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-97.0 + 97.0i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + (13.8 + 51.8i)T + (-6.85e3 + 3.96e3i)T^{2} \) |
| 97 | \( 1 + (-166. - 44.7i)T + (8.14e3 + 4.70e3i)T^{2} \) |
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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
−11.65279801672754857862314894947, −10.81069189150796312273149755681, −9.298790373207862068209437321300, −8.891644983136958198241723957358, −8.119545896636406545610211794326, −6.88056279040276617797551039902, −5.84408739900754915644164471683, −4.96495489631015609025486411729, −3.47477834424108485862018234969, −2.41961753300716257300868678805,
1.25061480972900784378421708451, 2.31388076263069118784494811366, 3.60850129457299795362072237484, 4.17867679511607217420436144453, 6.33828874952480928081810960142, 7.47840523737461314094472637290, 7.995433817133346628647279882971, 9.371511409866753985718133474011, 10.32743123932355984225161352241, 10.88058431713955920647605189388
