L(s) = 1 | + (0.909 + 0.415i)2-s + (0.654 + 0.755i)4-s + (−1.73 − 0.948i)5-s + (0.281 + 0.959i)8-s + (−1.18 − 1.58i)10-s + (0.156 − 0.469i)13-s + (−0.142 + 0.989i)16-s + (0.682 − 1.83i)17-s + (−0.420 − 1.93i)20-s + (1.57 + 2.45i)25-s + (0.337 − 0.362i)26-s + (1.53 − 1.23i)29-s + (−0.540 + 0.841i)32-s + (1.38 − 1.38i)34-s + (−0.521 − 1.25i)37-s + ⋯ |
L(s) = 1 | + (0.909 + 0.415i)2-s + (0.654 + 0.755i)4-s + (−1.73 − 0.948i)5-s + (0.281 + 0.959i)8-s + (−1.18 − 1.58i)10-s + (0.156 − 0.469i)13-s + (−0.142 + 0.989i)16-s + (0.682 − 1.83i)17-s + (−0.420 − 1.93i)20-s + (1.57 + 2.45i)25-s + (0.337 − 0.362i)26-s + (1.53 − 1.23i)29-s + (−0.540 + 0.841i)32-s + (1.38 − 1.38i)34-s + (−0.521 − 1.25i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3204 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3204 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.566908767\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.566908767\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.909 - 0.415i)T \) |
| 3 | \( 1 \) |
| 89 | \( 1 + (-0.479 - 0.877i)T \) |
good | 5 | \( 1 + (1.73 + 0.948i)T + (0.540 + 0.841i)T^{2} \) |
| 7 | \( 1 + (-0.977 - 0.212i)T^{2} \) |
| 11 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 13 | \( 1 + (-0.156 + 0.469i)T + (-0.800 - 0.599i)T^{2} \) |
| 17 | \( 1 + (-0.682 + 1.83i)T + (-0.755 - 0.654i)T^{2} \) |
| 19 | \( 1 + (-0.479 + 0.877i)T^{2} \) |
| 23 | \( 1 + (-0.877 - 0.479i)T^{2} \) |
| 29 | \( 1 + (-1.53 + 1.23i)T + (0.212 - 0.977i)T^{2} \) |
| 31 | \( 1 + (0.877 - 0.479i)T^{2} \) |
| 37 | \( 1 + (0.521 + 1.25i)T + (-0.707 + 0.707i)T^{2} \) |
| 41 | \( 1 + (0.199 + 0.599i)T + (-0.800 + 0.599i)T^{2} \) |
| 43 | \( 1 + (0.212 + 0.977i)T^{2} \) |
| 47 | \( 1 + (0.989 + 0.142i)T^{2} \) |
| 53 | \( 1 + (0.697 - 0.0498i)T + (0.989 - 0.142i)T^{2} \) |
| 59 | \( 1 + (-0.599 - 0.800i)T^{2} \) |
| 61 | \( 1 + (-1.63 + 1.13i)T + (0.349 - 0.936i)T^{2} \) |
| 67 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 71 | \( 1 + (0.540 - 0.841i)T^{2} \) |
| 73 | \( 1 + (1.29 + 0.186i)T + (0.959 + 0.281i)T^{2} \) |
| 79 | \( 1 + (-0.281 + 0.959i)T^{2} \) |
| 83 | \( 1 + (0.997 - 0.0713i)T^{2} \) |
| 97 | \( 1 + (-0.136 + 0.0401i)T + (0.841 - 0.540i)T^{2} \) |
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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
−8.529643119091996356742520929468, −7.79703284774437932205807854633, −7.46236492893711819479818854594, −6.62399233865640384267187333210, −5.41315409303762895191133242475, −4.96192821470580913234157652038, −4.16996085344610707893425132369, −3.51049327773065550805149891319, −2.62511907782419859564090393535, −0.77982380407643884635846703142,
1.37481286617886814875699030058, 2.80647505233360536920529339297, 3.43721401678607730022086587386, 4.08766516558694953942576468059, 4.74956029602286537791805831118, 5.92323436982774293930198857458, 6.70987194564085745461785631361, 7.15273375809045873906345184735, 8.138701899450947182024265229914, 8.616627152724284047210066941266