L(s) = 1 | + (−1.71 − 0.206i)3-s + 4.18·5-s + (2.91 + 0.711i)9-s − 1.42i·11-s + (−0.850 − 0.491i)13-s + (−7.19 − 0.866i)15-s + (−0.185 + 0.321i)17-s + (4.30 − 2.48i)19-s − 5.75i·23-s + 12.5·25-s + (−4.86 − 1.82i)27-s + (7.31 − 4.22i)29-s + (−6.28 + 3.62i)31-s + (−0.294 + 2.44i)33-s + (−1.73 − 3.00i)37-s + ⋯ |
L(s) = 1 | + (−0.992 − 0.119i)3-s + 1.87·5-s + (0.971 + 0.237i)9-s − 0.429i·11-s + (−0.235 − 0.136i)13-s + (−1.85 − 0.223i)15-s + (−0.0449 + 0.0779i)17-s + (0.988 − 0.570i)19-s − 1.20i·23-s + 2.50·25-s + (−0.936 − 0.351i)27-s + (1.35 − 0.784i)29-s + (−1.12 + 0.651i)31-s + (−0.0512 + 0.425i)33-s + (−0.284 − 0.493i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.606i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.794 + 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.825353384\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.825353384\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.71 + 0.206i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 4.18T + 5T^{2} \) |
| 11 | \( 1 + 1.42iT - 11T^{2} \) |
| 13 | \( 1 + (0.850 + 0.491i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (0.185 - 0.321i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.30 + 2.48i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 5.75iT - 23T^{2} \) |
| 29 | \( 1 + (-7.31 + 4.22i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (6.28 - 3.62i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.73 + 3.00i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.06 - 1.85i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.00 - 5.21i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4.13 - 7.16i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4.30 + 2.48i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.27 + 3.94i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (6.50 + 3.75i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.03 - 8.71i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 10.9iT - 71T^{2} \) |
| 73 | \( 1 + (-8.25 - 4.76i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.25 + 7.37i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.972 - 1.68i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-3.90 - 6.76i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (3.34 - 1.92i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.511400115211928962291666087634, −8.575316627603525328494312209375, −7.41163093651577720716701978060, −6.43694365734162863814387375340, −6.13842909783513149277868368901, −5.19700905593153562957152036669, −4.69760131817782120601826506075, −3.02725916003526216552166236600, −1.98525447101084030078269161331, −0.886313233136152138190020095856,
1.25533628024403701933132743221, 2.08456354377139491549679178558, 3.45783617399426438369474200962, 4.86820384721667728158537921566, 5.36414284787946895158545620484, 6.03774921798104214777632181435, 6.80013392737673349922437990953, 7.52898637926362133612737013066, 8.950480593807220501022097230232, 9.587936068774916674683912406545