L(s) = 1 | + (−1.24 − 0.678i)2-s + (1.07 + 1.68i)4-s + (0.920 + 0.920i)5-s + 4.02i·7-s + (−0.196 − 2.82i)8-s + (−0.517 − 1.76i)10-s + (−0.00588 − 0.00588i)11-s + (−1.05 + 1.05i)13-s + (2.72 − 4.98i)14-s + (−1.67 + 3.63i)16-s − 8.16·17-s + (−2.21 + 2.21i)19-s + (−0.556 + 2.54i)20-s + (0.00330 + 0.0112i)22-s + 4.29i·23-s + ⋯ |
L(s) = 1 | + (−0.877 − 0.479i)2-s + (0.539 + 0.841i)4-s + (0.411 + 0.411i)5-s + 1.51i·7-s + (−0.0695 − 0.997i)8-s + (−0.163 − 0.558i)10-s + (−0.00177 − 0.00177i)11-s + (−0.291 + 0.291i)13-s + (0.729 − 1.33i)14-s + (−0.417 + 0.908i)16-s − 1.97·17-s + (−0.509 + 0.509i)19-s + (−0.124 + 0.568i)20-s + (0.000705 + 0.00240i)22-s + 0.895i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0381 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0381 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.542450 + 0.522147i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.542450 + 0.522147i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.24 + 0.678i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.920 - 0.920i)T + 5iT^{2} \) |
| 7 | \( 1 - 4.02iT - 7T^{2} \) |
| 11 | \( 1 + (0.00588 + 0.00588i)T + 11iT^{2} \) |
| 13 | \( 1 + (1.05 - 1.05i)T - 13iT^{2} \) |
| 17 | \( 1 + 8.16T + 17T^{2} \) |
| 19 | \( 1 + (2.21 - 2.21i)T - 19iT^{2} \) |
| 23 | \( 1 - 4.29iT - 23T^{2} \) |
| 29 | \( 1 + (-1.74 + 1.74i)T - 29iT^{2} \) |
| 31 | \( 1 - 4.60T + 31T^{2} \) |
| 37 | \( 1 + (3.07 + 3.07i)T + 37iT^{2} \) |
| 41 | \( 1 - 9.43iT - 41T^{2} \) |
| 43 | \( 1 + (-6.55 - 6.55i)T + 43iT^{2} \) |
| 47 | \( 1 + 2.90T + 47T^{2} \) |
| 53 | \( 1 + (-8.65 - 8.65i)T + 53iT^{2} \) |
| 59 | \( 1 + (-6.98 - 6.98i)T + 59iT^{2} \) |
| 61 | \( 1 + (0.243 - 0.243i)T - 61iT^{2} \) |
| 67 | \( 1 + (-4.55 + 4.55i)T - 67iT^{2} \) |
| 71 | \( 1 + 10.8iT - 71T^{2} \) |
| 73 | \( 1 + 7.35iT - 73T^{2} \) |
| 79 | \( 1 - 8.69T + 79T^{2} \) |
| 83 | \( 1 + (8.93 - 8.93i)T - 83iT^{2} \) |
| 89 | \( 1 + 9.17iT - 89T^{2} \) |
| 97 | \( 1 - 3.27T + 97T^{2} \) |
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
−11.34039246142108884414035690124, −10.42619924695100793160415737357, −9.452580475539318441913330006198, −8.863860503893560297763949326875, −8.008044258891058871953422245651, −6.69077912075928922812322479602, −6.00883329214989407647347404062, −4.41545580201437926360685465446, −2.74940277731548208330006520904, −2.04942404739243027646550413514,
0.60624159323586641448093188216, 2.22640305346146703246380375765, 4.21943402148074830362072085186, 5.24288395925301206485049499699, 6.71259125812725714684148221402, 7.03527934183700921472481907170, 8.341223787121100881122646497661, 8.989179533258512884451558087629, 10.09133595089199123061064394628, 10.64307887084068403911870252211