L(s) = 1 | − 2.32i·5-s − 4.34·7-s − 0.264i·11-s + 6.38i·13-s + 2.56i·17-s + (−2.62 − 3.47i)19-s − 2.66i·23-s − 0.426·25-s + 8.92·29-s + 1.78i·31-s + 10.1i·35-s − 3.08i·37-s − 7.44·41-s + 5.05·43-s + 7.45i·47-s + ⋯ |
L(s) = 1 | − 1.04i·5-s − 1.64·7-s − 0.0797i·11-s + 1.76i·13-s + 0.621i·17-s + (−0.602 − 0.798i)19-s − 0.556i·23-s − 0.0852·25-s + 1.65·29-s + 0.320i·31-s + 1.71i·35-s − 0.507i·37-s − 1.16·41-s + 0.770·43-s + 1.08i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0310i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0310i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.222389101\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.222389101\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (2.62 + 3.47i)T \) |
good | 5 | \( 1 + 2.32iT - 5T^{2} \) |
| 7 | \( 1 + 4.34T + 7T^{2} \) |
| 11 | \( 1 + 0.264iT - 11T^{2} \) |
| 13 | \( 1 - 6.38iT - 13T^{2} \) |
| 17 | \( 1 - 2.56iT - 17T^{2} \) |
| 23 | \( 1 + 2.66iT - 23T^{2} \) |
| 29 | \( 1 - 8.92T + 29T^{2} \) |
| 31 | \( 1 - 1.78iT - 31T^{2} \) |
| 37 | \( 1 + 3.08iT - 37T^{2} \) |
| 41 | \( 1 + 7.44T + 41T^{2} \) |
| 43 | \( 1 - 5.05T + 43T^{2} \) |
| 47 | \( 1 - 7.45iT - 47T^{2} \) |
| 53 | \( 1 - 4.10T + 53T^{2} \) |
| 59 | \( 1 - 8.36T + 59T^{2} \) |
| 61 | \( 1 + 8.45T + 61T^{2} \) |
| 67 | \( 1 - 2.35iT - 67T^{2} \) |
| 71 | \( 1 + 5.02T + 71T^{2} \) |
| 73 | \( 1 - 14.9T + 73T^{2} \) |
| 79 | \( 1 + 6.23iT - 79T^{2} \) |
| 83 | \( 1 - 16.0iT - 83T^{2} \) |
| 89 | \( 1 - 8.25T + 89T^{2} \) |
| 97 | \( 1 + 15.1iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.867792564601474528985838759062, −8.403880382401392342207228820220, −7.02448222631768992887590140276, −6.59899419128853149729351030964, −5.90956205910014349269203132808, −4.70291604758952722776352891955, −4.22574265015476693678313698671, −3.16589957872834171224732966038, −2.10382009832878025367634973524, −0.75219107865496680305110247954,
0.59174316994332836852671077856, 2.46686424274784030374222506856, 3.16240661415087748816052868353, 3.65114503787719019198442212211, 5.04081282328611217586663167648, 5.97602609581574671597572386515, 6.50424276167379447054815611635, 7.18887237004807573608540638486, 7.978354811411018515834225495700, 8.835889956066646458029244942637