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 | \( 1 - 3.27T + 97T^{2} \) |
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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
−10.64307887084068403911870252211, −10.09133595089199123061064394628, −8.989179533258512884451558087629, −8.341223787121100881122646497661, −7.03527934183700921472481907170, −6.71259125812725714684148221402, −5.24288395925301206485049499699, −4.21943402148074830362072085186, −2.22640305346146703246380375765, −0.60624159323586641448093188216,
2.04942404739243027646550413514, 2.74940277731548208330006520904, 4.41545580201437926360685465446, 6.00883329214989407647347404062, 6.69077912075928922812322479602, 8.008044258891058871953422245651, 8.863860503893560297763949326875, 9.452580475539318441913330006198, 10.42619924695100793160415737357, 11.34039246142108884414035690124