L(s) = 1 | + (0.275 − 0.158i)5-s + (−1.25 − 0.724i)7-s + (−0.548 + 0.949i)11-s + (−1.44 − 2.51i)13-s − 3.46i·17-s + 4.89i·19-s + (1.41 + 2.44i)23-s + (−2.44 + 4.24i)25-s + (−7.89 − 4.56i)29-s + (6.45 − 3.72i)31-s − 0.460·35-s + 4.89·37-s + (−8.44 + 4.87i)41-s + (−5.97 − 3.44i)43-s + (−4.56 + 7.89i)47-s + ⋯ |
L(s) = 1 | + (0.123 − 0.0710i)5-s + (−0.474 − 0.273i)7-s + (−0.165 + 0.286i)11-s + (−0.402 − 0.696i)13-s − 0.840i·17-s + 1.12i·19-s + (0.294 + 0.510i)23-s + (−0.489 + 0.848i)25-s + (−1.46 − 0.846i)29-s + (1.15 − 0.668i)31-s − 0.0778·35-s + 0.805·37-s + (−1.31 + 0.761i)41-s + (−0.911 − 0.526i)43-s + (−0.665 + 1.15i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.906 - 0.422i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.906 - 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2090166099\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2090166099\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.275 + 0.158i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (1.25 + 0.724i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.548 - 0.949i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.44 + 2.51i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 3.46iT - 17T^{2} \) |
| 19 | \( 1 - 4.89iT - 19T^{2} \) |
| 23 | \( 1 + (-1.41 - 2.44i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (7.89 + 4.56i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-6.45 + 3.72i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 4.89T + 37T^{2} \) |
| 41 | \( 1 + (8.44 - 4.87i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5.97 + 3.44i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4.56 - 7.89i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 4.41iT - 53T^{2} \) |
| 59 | \( 1 + (-4.56 - 7.89i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2 - 3.46i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.41 - 2.55i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 7.56T + 71T^{2} \) |
| 73 | \( 1 - 1.89T + 73T^{2} \) |
| 79 | \( 1 + (-1.73 - i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (7.93 - 13.7i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 11.9iT - 89T^{2} \) |
| 97 | \( 1 + (-2.5 + 4.33i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.530943417527224239769986057113, −8.309510445333311531400021693294, −7.66804725637665733853024415962, −7.04675393084562095798914634633, −6.01697351589339936146500884146, −5.42481672753802500275518816003, −4.45832995950294297180226915411, −3.51383258288459783778970509195, −2.66735245335462995189307142728, −1.43247724401745467803188392050,
0.06602528640683071258317304624, 1.72988423699714022868361600197, 2.72729507804005166323765089301, 3.62838229266459428246985932833, 4.65637292415554858248601357495, 5.38691921419832954116152532473, 6.50431548913372609941660121997, 6.73559036795064970655659073695, 7.88158920378172006837965059740, 8.617592620480385572147814405790