L(s) = 1 | + (−0.5 − 0.866i)3-s + (0.751 − 1.30i)5-s + (−2.44 − 1.00i)7-s + (−0.499 + 0.866i)9-s + (−0.5 − 0.866i)11-s − 3.15·13-s − 1.50·15-s + (0.761 + 1.31i)17-s + (−2.84 + 4.92i)19-s + (0.356 + 2.62i)21-s + (−2.70 + 4.67i)23-s + (1.36 + 2.37i)25-s + 0.999·27-s − 0.972·29-s + (1.57 + 2.73i)31-s + ⋯ |
L(s) = 1 | + (−0.288 − 0.499i)3-s + (0.336 − 0.582i)5-s + (−0.925 − 0.378i)7-s + (−0.166 + 0.288i)9-s + (−0.150 − 0.261i)11-s − 0.875·13-s − 0.388·15-s + (0.184 + 0.320i)17-s + (−0.652 + 1.13i)19-s + (0.0777 + 0.572i)21-s + (−0.563 + 0.975i)23-s + (0.273 + 0.474i)25-s + 0.192·27-s − 0.180·29-s + (0.283 + 0.491i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.319 - 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.319 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6498225669\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6498225669\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (2.44 + 1.00i)T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
good | 5 | \( 1 + (-0.751 + 1.30i)T + (-2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 + 3.15T + 13T^{2} \) |
| 17 | \( 1 + (-0.761 - 1.31i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.84 - 4.92i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.70 - 4.67i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 0.972T + 29T^{2} \) |
| 31 | \( 1 + (-1.57 - 2.73i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.40 + 5.90i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 + 4.60T + 43T^{2} \) |
| 47 | \( 1 + (-3.58 + 6.20i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.40 - 4.16i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.35 - 5.80i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.64 - 9.78i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.58 - 2.74i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 14.9T + 71T^{2} \) |
| 73 | \( 1 + (3.74 + 6.48i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.07 + 3.59i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 6.21T + 83T^{2} \) |
| 89 | \( 1 + (2.34 - 4.06i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 16.2T + 97T^{2} \) |
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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
−9.400039265993406267233546500414, −8.676302206505829683053947931168, −7.63746317401603196753078612559, −7.16215989142872149335342446028, −5.96070020111992574930636116737, −5.71735758036934815086756007184, −4.48834054843219750570732406209, −3.54087332686248179182044249008, −2.35721433617301027585614717032, −1.17075675309885809318640722983,
0.26227808456182913120575017068, 2.42362082308313400676037813854, 2.90001092208094026913255093135, 4.23016892965427194875540083614, 4.94969461154756680519973484007, 6.05434462368530895715962523848, 6.53553023729140982180012071012, 7.35622724039550201469108137723, 8.433640071750200726405864511205, 9.340197201645978157441539985888