L(s) = 1 | + (−0.712 − 1.57i)3-s + (1.02 + 0.594i)5-s − 1.04·7-s + (−1.98 + 2.24i)9-s − 3.41i·11-s + (0.793 − 0.458i)13-s + (0.205 − 2.04i)15-s + (−2.59 − 1.49i)17-s + (−4.30 − 0.688i)19-s + (0.744 + 1.65i)21-s + (−1.62 + 0.940i)23-s + (−1.79 − 3.10i)25-s + (4.96 + 1.53i)27-s + (0.797 + 1.38i)29-s − 8.98i·31-s + ⋯ |
L(s) = 1 | + (−0.411 − 0.911i)3-s + (0.460 + 0.265i)5-s − 0.395·7-s + (−0.662 + 0.749i)9-s − 1.02i·11-s + (0.220 − 0.127i)13-s + (0.0530 − 0.529i)15-s + (−0.628 − 0.363i)17-s + (−0.987 − 0.157i)19-s + (0.162 + 0.360i)21-s + (−0.339 + 0.196i)23-s + (−0.358 − 0.621i)25-s + (0.955 + 0.295i)27-s + (0.148 + 0.256i)29-s − 1.61i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 + 0.354i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.934 + 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.131268 - 0.715477i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.131268 - 0.715477i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.712 + 1.57i)T \) |
| 19 | \( 1 + (4.30 + 0.688i)T \) |
good | 5 | \( 1 + (-1.02 - 0.594i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + 1.04T + 7T^{2} \) |
| 11 | \( 1 + 3.41iT - 11T^{2} \) |
| 13 | \( 1 + (-0.793 + 0.458i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2.59 + 1.49i)T + (8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (1.62 - 0.940i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.797 - 1.38i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 8.98iT - 31T^{2} \) |
| 37 | \( 1 + 4.54iT - 37T^{2} \) |
| 41 | \( 1 + (0.469 - 0.813i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.73 + 4.73i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (7.82 - 4.51i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.0418 - 0.0725i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (5.12 - 8.87i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.766 + 1.32i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (8.67 - 5.01i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (1.80 - 3.13i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.38 + 7.58i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.95 - 1.70i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 7.04iT - 83T^{2} \) |
| 89 | \( 1 + (2.48 + 4.31i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (7.48 + 4.32i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.753923036708208647504949955991, −8.755970566623143478511028389633, −8.017986065918161946891125125905, −7.02357907371800589416034780221, −6.14411744580016139659878007394, −5.80328949497280361682199860561, −4.39095466197766168915924001499, −2.98275655596944376388081338382, −1.98332182366877589022990121446, −0.34179327058085414914310807954,
1.83351940434828350892784113658, 3.29756400814093123837116114442, 4.38654097635066787110300594148, 5.05126902429586797991533838000, 6.18694643575730906966675806715, 6.74064762951031877408616196035, 8.153372542127940289684435417337, 9.024218669637384928289478402906, 9.703699340212452644100863293142, 10.35176760190306788774612748923