| L(s) = 1 | + (−1.08 + 0.419i)2-s + (0.547 + 1.64i)3-s + (−0.479 + 0.437i)4-s + (−0.585 + 2.05i)5-s + (−1.28 − 1.55i)6-s + (−2.43 + 3.66i)7-s + (1.37 − 2.75i)8-s + (−2.39 + 1.80i)9-s + (−0.229 − 2.47i)10-s + (2.49 + 2.01i)11-s + (−0.981 − 0.548i)12-s + (0.268 − 0.405i)13-s + (1.09 − 4.99i)14-s + (−3.69 + 0.165i)15-s + (−0.210 + 2.27i)16-s + (−4.44 + 2.38i)17-s + ⋯ |
| L(s) = 1 | + (−0.766 + 0.296i)2-s + (0.316 + 0.948i)3-s + (−0.239 + 0.218i)4-s + (−0.261 + 0.919i)5-s + (−0.524 − 0.633i)6-s + (−0.918 + 1.38i)7-s + (0.485 − 0.974i)8-s + (−0.799 + 0.600i)9-s + (−0.0725 − 0.782i)10-s + (0.753 + 0.606i)11-s + (−0.283 − 0.158i)12-s + (0.0744 − 0.112i)13-s + (0.292 − 1.33i)14-s + (−0.955 + 0.0426i)15-s + (−0.0525 + 0.567i)16-s + (−1.07 + 0.578i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.442 + 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.442 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.378640 - 0.609200i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.378640 - 0.609200i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.547 - 1.64i)T \) |
| 103 | \( 1 + (-1.03 + 10.0i)T \) |
| good | 2 | \( 1 + (1.08 - 0.419i)T + (1.47 - 1.34i)T^{2} \) |
| 5 | \( 1 + (0.585 - 2.05i)T + (-4.25 - 2.63i)T^{2} \) |
| 7 | \( 1 + (2.43 - 3.66i)T + (-2.72 - 6.44i)T^{2} \) |
| 11 | \( 1 + (-2.49 - 2.01i)T + (2.35 + 10.7i)T^{2} \) |
| 13 | \( 1 + (-0.268 + 0.405i)T + (-5.06 - 11.9i)T^{2} \) |
| 17 | \( 1 + (4.44 - 2.38i)T + (9.39 - 14.1i)T^{2} \) |
| 19 | \( 1 + (-0.994 - 2.82i)T + (-14.8 + 11.9i)T^{2} \) |
| 23 | \( 1 + (-1.35 - 8.75i)T + (-21.9 + 6.97i)T^{2} \) |
| 29 | \( 1 + (-2.00 + 7.04i)T + (-24.6 - 15.2i)T^{2} \) |
| 31 | \( 1 + (-1.91 + 0.883i)T + (20.1 - 23.5i)T^{2} \) |
| 37 | \( 1 + (-1.12 + 1.49i)T + (-10.1 - 35.5i)T^{2} \) |
| 41 | \( 1 + (-8.61 - 2.16i)T + (36.1 + 19.3i)T^{2} \) |
| 43 | \( 1 + (7.37 + 9.76i)T + (-11.7 + 41.3i)T^{2} \) |
| 47 | \( 1 + (-3.89 - 6.74i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.00 + 7.01i)T + (-8.12 - 52.3i)T^{2} \) |
| 59 | \( 1 + (-8.35 - 12.6i)T + (-22.9 + 54.3i)T^{2} \) |
| 61 | \( 1 + (-10.5 + 5.66i)T + (33.6 - 50.8i)T^{2} \) |
| 67 | \( 1 + (1.50 - 3.02i)T + (-40.3 - 53.4i)T^{2} \) |
| 71 | \( 1 + (-0.208 - 0.0523i)T + (62.5 + 33.5i)T^{2} \) |
| 73 | \( 1 + (0.848 + 2.98i)T + (-62.0 + 38.4i)T^{2} \) |
| 79 | \( 1 + (-2.09 - 2.16i)T + (-2.43 + 78.9i)T^{2} \) |
| 83 | \( 1 + (2.31 + 4.64i)T + (-50.0 + 66.2i)T^{2} \) |
| 89 | \( 1 + (-9.95 - 9.07i)T + (8.21 + 88.6i)T^{2} \) |
| 97 | \( 1 + (-2.97 + 1.84i)T + (43.2 - 86.8i)T^{2} \) |
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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
−10.19334413729546603070270416712, −9.659782415849092646439590722004, −9.067723789411189728495873245871, −8.404286727540969811210426520222, −7.39926874815452205220835225595, −6.48602938552324475566504816348, −5.57277400860085466852803814727, −4.13324314522087291403938055460, −3.47064625297240471857094550133, −2.37022067831028683929641318494,
0.54008645569499931633214581544, 1.03076267826718333862024141495, 2.68063381867335526923268829015, 4.04932162482308920969206388343, 4.96702365204498071726142090501, 6.49603702298167802526858947736, 6.92821418489892564263626099573, 8.077463019422037931061444022260, 8.864186282310782012691102227478, 9.151440469572519177772014340895
