| L(s) = 1 | + (1.38 + 1.00i)2-s + (0.309 + 0.951i)3-s + (0.281 + 0.867i)4-s + (−1.02 − 1.98i)5-s + (−0.527 + 1.62i)6-s − 3.94·7-s + (0.573 − 1.76i)8-s + (−0.809 + 0.587i)9-s + (0.573 − 3.77i)10-s + (4.78 + 3.47i)11-s + (−0.737 + 0.535i)12-s + (−2.66 + 1.93i)13-s + (−5.44 − 3.95i)14-s + (1.57 − 1.59i)15-s + (4.03 − 2.93i)16-s + (0.836 − 2.57i)17-s + ⋯ |
| L(s) = 1 | + (0.976 + 0.709i)2-s + (0.178 + 0.549i)3-s + (0.140 + 0.433i)4-s + (−0.459 − 0.888i)5-s + (−0.215 + 0.662i)6-s − 1.49·7-s + (0.202 − 0.624i)8-s + (−0.269 + 0.195i)9-s + (0.181 − 1.19i)10-s + (1.44 + 1.04i)11-s + (−0.212 + 0.154i)12-s + (−0.738 + 0.536i)13-s + (−1.45 − 1.05i)14-s + (0.405 − 0.410i)15-s + (1.00 − 0.733i)16-s + (0.202 − 0.624i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.675 - 0.737i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.675 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.20782 + 0.531449i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20782 + 0.531449i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.309 - 0.951i)T \) |
| 5 | \( 1 + (1.02 + 1.98i)T \) |
| good | 2 | \( 1 + (-1.38 - 1.00i)T + (0.618 + 1.90i)T^{2} \) |
| 7 | \( 1 + 3.94T + 7T^{2} \) |
| 11 | \( 1 + (-4.78 - 3.47i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (2.66 - 1.93i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.836 + 2.57i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (0.728 - 2.24i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-0.472 - 0.343i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (1.20 + 3.72i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-0.837 + 2.57i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-0.0168 + 0.0122i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (1.19 - 0.865i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 1.27T + 43T^{2} \) |
| 47 | \( 1 + (-1.67 - 5.16i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-0.870 - 2.67i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-3.79 + 2.75i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (4.51 + 3.28i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-1.86 + 5.73i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (2.50 + 7.70i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (10.7 + 7.82i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-5.14 - 15.8i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-0.241 + 0.743i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-2.80 - 2.04i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (0.758 + 2.33i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−14.80460016060611820747030094494, −13.76486616486753639542074291225, −12.63787418127742219187503894248, −11.96747231879242556697590869665, −9.755072778896576711686205875053, −9.341786533315010174947218659284, −7.34333683891994934371985838875, −6.20995597231840924001267324057, −4.68547529185890976712171486678, −3.77890393163148542464647578407,
2.92892643071426576466133044833, 3.74855643477579559495475683320, 6.01136285493052456653856731428, 7.07441591885586723161032691744, 8.708240694109296259293916199803, 10.29897104248099282809299647481, 11.48789565158508524841065236416, 12.33361798543891736870521559756, 13.21056808063903917057686144838, 14.18286110050608806298946740377
