L(s) = 1 | + (0.995 − 0.0950i)3-s + (0.786 + 0.618i)4-s + (−0.654 + 0.755i)7-s + (0.981 − 0.189i)9-s + (0.841 + 0.540i)12-s + (−1.16 + 0.600i)13-s + (0.235 + 0.971i)16-s + (−0.0359 + 0.186i)19-s + (−0.580 + 0.814i)21-s + (0.0475 − 0.998i)25-s + (0.959 − 0.281i)27-s + (−0.981 + 0.189i)28-s + (−0.0913 − 1.91i)31-s + (0.888 + 0.458i)36-s + (−0.0475 − 0.0824i)37-s + ⋯ |
L(s) = 1 | + (0.995 − 0.0950i)3-s + (0.786 + 0.618i)4-s + (−0.654 + 0.755i)7-s + (0.981 − 0.189i)9-s + (0.841 + 0.540i)12-s + (−1.16 + 0.600i)13-s + (0.235 + 0.971i)16-s + (−0.0359 + 0.186i)19-s + (−0.580 + 0.814i)21-s + (0.0475 − 0.998i)25-s + (0.959 − 0.281i)27-s + (−0.981 + 0.189i)28-s + (−0.0913 − 1.91i)31-s + (0.888 + 0.458i)36-s + (−0.0475 − 0.0824i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.647879710\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.647879710\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.995 + 0.0950i)T \) |
| 7 | \( 1 + (0.654 - 0.755i)T \) |
| 67 | \( 1 + (0.841 + 0.540i)T \) |
good | 2 | \( 1 + (-0.786 - 0.618i)T^{2} \) |
| 5 | \( 1 + (-0.0475 + 0.998i)T^{2} \) |
| 11 | \( 1 + (0.0475 - 0.998i)T^{2} \) |
| 13 | \( 1 + (1.16 - 0.600i)T + (0.580 - 0.814i)T^{2} \) |
| 17 | \( 1 + (-0.959 + 0.281i)T^{2} \) |
| 19 | \( 1 + (0.0359 - 0.186i)T + (-0.928 - 0.371i)T^{2} \) |
| 23 | \( 1 + (-0.981 + 0.189i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.0913 + 1.91i)T + (-0.995 + 0.0950i)T^{2} \) |
| 37 | \( 1 + (0.0475 + 0.0824i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.235 + 0.971i)T^{2} \) |
| 43 | \( 1 + (-1.07 + 0.153i)T + (0.959 - 0.281i)T^{2} \) |
| 47 | \( 1 + (-0.654 - 0.755i)T^{2} \) |
| 53 | \( 1 + (0.723 - 0.690i)T^{2} \) |
| 59 | \( 1 + (0.580 + 0.814i)T^{2} \) |
| 61 | \( 1 + (1.44 + 1.37i)T + (0.0475 + 0.998i)T^{2} \) |
| 71 | \( 1 + (-0.723 + 0.690i)T^{2} \) |
| 73 | \( 1 + (-0.547 - 1.86i)T + (-0.841 + 0.540i)T^{2} \) |
| 79 | \( 1 + (-1.08 + 1.68i)T + (-0.415 - 0.909i)T^{2} \) |
| 83 | \( 1 + (0.0475 - 0.998i)T^{2} \) |
| 89 | \( 1 + (0.981 + 0.189i)T^{2} \) |
| 97 | \( 1 + (0.580 - 1.00i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−9.607460331281947776670963164834, −9.095769054319113263870577385156, −8.112251184285836074577967561584, −7.54899348788613932221918225185, −6.71537110816757238642503123265, −5.98956298133176529115311700651, −4.54328419358289065296078367147, −3.60288900835790033981760824491, −2.58306362419774008749270651687, −2.12307633189289024427545966301,
1.37546578361237762968400389448, 2.66720672491230161110063970927, 3.30879966185866007335839445360, 4.54534471068192349794495474681, 5.49300803877584302264964911186, 6.69770449239182493162363578505, 7.24126851972595454617822223075, 7.84609399978797304495257370204, 9.085067960272526365523432632529, 9.652768556992512494965159666223