| L(s) = 1 | + (1.15 − 0.810i)2-s + (0.453 + 0.891i)3-s + (0.687 − 1.87i)4-s + (1.99 − 1.01i)5-s + (1.24 + 0.665i)6-s + (−2.49 − 2.49i)7-s + (−0.724 − 2.73i)8-s + (−0.587 + 0.809i)9-s + (1.48 − 2.79i)10-s + (−0.801 − 1.10i)11-s + (1.98 − 0.239i)12-s + (0.636 + 4.02i)13-s + (−4.90 − 0.869i)14-s + (1.81 + 1.31i)15-s + (−3.05 − 2.58i)16-s + (5.08 + 2.58i)17-s + ⋯ |
| L(s) = 1 | + (0.819 − 0.572i)2-s + (0.262 + 0.514i)3-s + (0.343 − 0.939i)4-s + (0.890 − 0.455i)5-s + (0.509 + 0.271i)6-s + (−0.941 − 0.941i)7-s + (−0.256 − 0.966i)8-s + (−0.195 + 0.269i)9-s + (0.468 − 0.883i)10-s + (−0.241 − 0.332i)11-s + (0.573 − 0.0692i)12-s + (0.176 + 1.11i)13-s + (−1.31 − 0.232i)14-s + (0.467 + 0.338i)15-s + (−0.763 − 0.645i)16-s + (1.23 + 0.627i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.533 + 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.533 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.95845 - 1.07957i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.95845 - 1.07957i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.15 + 0.810i)T \) |
| 3 | \( 1 + (-0.453 - 0.891i)T \) |
| 5 | \( 1 + (-1.99 + 1.01i)T \) |
| good | 7 | \( 1 + (2.49 + 2.49i)T + 7iT^{2} \) |
| 11 | \( 1 + (0.801 + 1.10i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.636 - 4.02i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (-5.08 - 2.58i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (-1.33 - 4.10i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (1.11 - 7.06i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (8.75 + 2.84i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-3.06 + 0.995i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-0.129 + 0.0204i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (-3.36 - 2.44i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (-5.22 + 5.22i)T - 43iT^{2} \) |
| 47 | \( 1 + (7.83 - 3.99i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (4.72 - 2.40i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (2.82 + 2.04i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-2.94 + 2.14i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (2.58 - 5.06i)T + (-39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (1.20 + 0.391i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (3.00 + 0.475i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (-4.80 + 14.7i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-4.03 - 2.05i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (8.38 + 11.5i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-7.56 - 14.8i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−11.63037128719057767953043513143, −10.54858913867341066369024882196, −9.776702548657522380645166340287, −9.358342267689244698928873463603, −7.64499871387695670287566544883, −6.23481066924079265984992435076, −5.53461137310507384491504808778, −4.12914308730965956827949688164, −3.33604385779719447039991850387, −1.60686484656783257710613861835,
2.53780626002364283288545743136, 3.18419641968317621495737353316, 5.21480720720879885569423927115, 5.94715596406241270599633426732, 6.81377268518320600218908357944, 7.78999875589047262495643783761, 8.967854800611753062000134127204, 9.889968634278808354826251802323, 11.15195305077387942925609468756, 12.47206788981165031376908944405
