| L(s) = 1 | + (0.318 + 0.267i)2-s + (−0.159 − 1.72i)3-s + (−0.317 − 1.79i)4-s + (0.409 − 0.591i)6-s + (0.229 − 1.29i)7-s + (0.795 − 1.37i)8-s + (−2.94 + 0.551i)9-s + (4.90 − 1.78i)11-s + (−3.05 + 0.834i)12-s + (0.0138 − 0.0116i)13-s + (0.419 − 0.352i)14-s + (−2.81 + 1.02i)16-s + (−1.56 − 2.71i)17-s + (−1.08 − 0.612i)18-s + (−0.208 + 0.361i)19-s + ⋯ |
| L(s) = 1 | + (0.225 + 0.188i)2-s + (−0.0922 − 0.995i)3-s + (−0.158 − 0.899i)4-s + (0.167 − 0.241i)6-s + (0.0866 − 0.491i)7-s + (0.281 − 0.486i)8-s + (−0.982 + 0.183i)9-s + (1.47 − 0.537i)11-s + (−0.881 + 0.241i)12-s + (0.00383 − 0.00321i)13-s + (0.112 − 0.0941i)14-s + (−0.703 + 0.256i)16-s + (−0.379 − 0.658i)17-s + (−0.255 − 0.144i)18-s + (−0.0478 + 0.0829i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.736 + 0.676i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.736 + 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.526185 - 1.35136i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.526185 - 1.35136i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.159 + 1.72i)T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + (-0.318 - 0.267i)T + (0.347 + 1.96i)T^{2} \) |
| 7 | \( 1 + (-0.229 + 1.29i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (-4.90 + 1.78i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.0138 + 0.0116i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (1.56 + 2.71i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.208 - 0.361i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.179 - 1.01i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (5.98 + 5.01i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.647 - 3.67i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-2.21 - 3.83i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.81 - 2.36i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (7.80 - 2.84i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.23 + 6.99i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 1.30T + 53T^{2} \) |
| 59 | \( 1 + (-3.47 - 1.26i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.20 + 6.80i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-8.44 + 7.08i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-3.04 - 5.26i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (0.273 - 0.473i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.374 - 0.314i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (3.53 + 2.96i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-1.68 + 2.92i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-9.34 + 3.40i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.14751668783970845519628576095, −9.260443139134693426371122439164, −8.431020474945202536338984132347, −7.22872065819903089238414300040, −6.58948285514967127893009252797, −5.87205684951414267636878342994, −4.79734081651286234769074467305, −3.57758720805954885249524263219, −1.86222458672926423336553022083, −0.75220362470651140699437670473,
2.19588921289253512907295279318, 3.57491424097319745816096588908, 4.14460962669847745671606901376, 5.13777903888452829177494697286, 6.27957913307692559685186829930, 7.35711852094336720378873776941, 8.622524780432419390697792374749, 8.982490069268780253032083174718, 9.887036133675585424427245528359, 11.00717936246322104753323230946
