L(s) = 1 | + (1.5 − 0.866i)5-s + (−1.80 + 3.12i)7-s + (−0.635 − 0.367i)11-s + (0.527 − 0.304i)13-s + 5.52·17-s − 2i·19-s + (−2.36 − 4.10i)23-s + (−1 + 1.73i)25-s + (6.78 + 3.91i)29-s + (4.70 + 8.15i)31-s + 6.25i·35-s − 2.34i·37-s + (−4.26 − 7.38i)41-s + (8.88 + 5.12i)43-s + (−5.88 + 10.1i)47-s + ⋯ |
L(s) = 1 | + (0.670 − 0.387i)5-s + (−0.682 + 1.18i)7-s + (−0.191 − 0.110i)11-s + (0.146 − 0.0845i)13-s + 1.33·17-s − 0.458i·19-s + (−0.493 − 0.855i)23-s + (−0.200 + 0.346i)25-s + (1.26 + 0.727i)29-s + (0.845 + 1.46i)31-s + 1.05i·35-s − 0.384i·37-s + (−0.665 − 1.15i)41-s + (1.35 + 0.782i)43-s + (−0.857 + 1.48i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.742 - 0.669i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.742 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.780251220\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.780251220\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-1.5 + 0.866i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (1.80 - 3.12i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.635 + 0.367i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.527 + 0.304i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 5.52T + 17T^{2} \) |
| 19 | \( 1 + 2iT - 19T^{2} \) |
| 23 | \( 1 + (2.36 + 4.10i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-6.78 - 3.91i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.70 - 8.15i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 2.34iT - 37T^{2} \) |
| 41 | \( 1 + (4.26 + 7.38i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-8.88 - 5.12i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (5.88 - 10.1i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 13.0iT - 53T^{2} \) |
| 59 | \( 1 + (1.04 - 0.604i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-9.78 - 5.65i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.46 + 3.15i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 2.63T + 71T^{2} \) |
| 73 | \( 1 + 2.05T + 73T^{2} \) |
| 79 | \( 1 + (1.24 - 2.15i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-10.6 - 6.12i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 1.94T + 89T^{2} \) |
| 97 | \( 1 + (-7.78 + 13.4i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.324300615039796420123087848662, −8.771124790012654148350667090547, −8.009956943283714105839920087633, −6.88050662230835292283888616138, −6.02207472514254415394214248562, −5.51639982626316865068709555613, −4.63005017359721214233872301665, −3.21239956619940656950488001428, −2.54521702014093961591703571450, −1.18738949407177946112074723341,
0.77373554605918909965692336427, 2.16466350155324272439538526679, 3.34311316659263361963122839582, 4.04185233228283877043812854259, 5.23035779857015164697686593018, 6.18053419985837898139528484242, 6.69079822233079387203942858348, 7.70944568402205313721131225921, 8.220740345241425318011206818594, 9.663497094875264747548407815067