L(s) = 1 | + 2-s + (−1.68 + 0.392i)3-s + 4-s + 5-s + (−1.68 + 0.392i)6-s − 2.72i·7-s + 8-s + (2.69 − 1.32i)9-s + 10-s + 5.40·11-s + (−1.68 + 0.392i)12-s + 1.90i·13-s − 2.72i·14-s + (−1.68 + 0.392i)15-s + 16-s − 1.89i·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.974 + 0.226i)3-s + 0.5·4-s + 0.447·5-s + (−0.688 + 0.160i)6-s − 1.02i·7-s + 0.353·8-s + (0.897 − 0.441i)9-s + 0.316·10-s + 1.62·11-s + (−0.487 + 0.113i)12-s + 0.527i·13-s − 0.727i·14-s + (−0.435 + 0.101i)15-s + 0.250·16-s − 0.459i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.798 + 0.602i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.798 + 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.382157057\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.382157057\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (1.68 - 0.392i)T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + (-5.24 - 6.28i)T \) |
good | 7 | \( 1 + 2.72iT - 7T^{2} \) |
| 11 | \( 1 - 5.40T + 11T^{2} \) |
| 13 | \( 1 - 1.90iT - 13T^{2} \) |
| 17 | \( 1 + 1.89iT - 17T^{2} \) |
| 19 | \( 1 + 2.89T + 19T^{2} \) |
| 23 | \( 1 - 0.0386iT - 23T^{2} \) |
| 29 | \( 1 + 9.54iT - 29T^{2} \) |
| 31 | \( 1 - 0.791iT - 31T^{2} \) |
| 37 | \( 1 + 4.33T + 37T^{2} \) |
| 41 | \( 1 - 2.80T + 41T^{2} \) |
| 43 | \( 1 - 9.14iT - 43T^{2} \) |
| 47 | \( 1 + 3.02iT - 47T^{2} \) |
| 53 | \( 1 - 3.05T + 53T^{2} \) |
| 59 | \( 1 + 8.27iT - 59T^{2} \) |
| 61 | \( 1 + 14.8iT - 61T^{2} \) |
| 71 | \( 1 + 14.3iT - 71T^{2} \) |
| 73 | \( 1 - 11.1T + 73T^{2} \) |
| 79 | \( 1 - 15.7iT - 79T^{2} \) |
| 83 | \( 1 - 4.38iT - 83T^{2} \) |
| 89 | \( 1 - 13.1iT - 89T^{2} \) |
| 97 | \( 1 - 5.86iT - 97T^{2} \) |
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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
−9.553150947455022321477502864390, −8.163635473954869016555845047777, −7.00327344360354413028710145446, −6.61263707454358487700503640595, −5.99186697411692488326894685903, −4.91239848475884657064041340186, −4.20556173691621772618847881962, −3.65863535160105866387642632737, −1.98510914701410649482728774205, −0.867285713828334963727677771807,
1.29397863275379874202854513510, 2.23020856063094860212063796699, 3.55622582395708003888303722339, 4.48432685906168241093347119952, 5.45225664984354642083913715974, 5.90407628363482168592199651762, 6.63884382492037609165740833207, 7.26868348131263114713018629367, 8.607969099697363732897293022561, 9.139364516427861572304095505622