L(s) = 1 | + (−0.467 + 0.883i)2-s + (1.67 + 0.216i)3-s + (−0.562 − 0.826i)4-s + (−0.972 + 1.37i)6-s + (0.993 − 0.110i)8-s + (1.77 + 0.467i)9-s + (−1.33 + 0.924i)11-s + (−0.760 − 1.50i)12-s + (−0.367 + 0.929i)16-s + (1.69 + 0.125i)17-s + (−1.24 + 1.35i)18-s + (−0.531 + 0.847i)19-s + (−0.193 − 1.61i)22-s + (1.68 + 0.0309i)24-s + (0.957 − 0.289i)25-s + ⋯ |
L(s) = 1 | + (−0.467 + 0.883i)2-s + (1.67 + 0.216i)3-s + (−0.562 − 0.826i)4-s + (−0.972 + 1.37i)6-s + (0.993 − 0.110i)8-s + (1.77 + 0.467i)9-s + (−1.33 + 0.924i)11-s + (−0.760 − 1.50i)12-s + (−0.367 + 0.929i)16-s + (1.69 + 0.125i)17-s + (−1.24 + 1.35i)18-s + (−0.531 + 0.847i)19-s + (−0.193 − 1.61i)22-s + (1.68 + 0.0309i)24-s + (0.957 − 0.289i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0410 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0410 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.621774693\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.621774693\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.467 - 0.883i)T \) |
| 19 | \( 1 + (0.531 - 0.847i)T \) |
good | 3 | \( 1 + (-1.67 - 0.216i)T + (0.967 + 0.254i)T^{2} \) |
| 5 | \( 1 + (-0.957 + 0.289i)T^{2} \) |
| 7 | \( 1 + (-0.0275 - 0.999i)T^{2} \) |
| 11 | \( 1 + (1.33 - 0.924i)T + (0.350 - 0.936i)T^{2} \) |
| 13 | \( 1 + (0.995 - 0.0917i)T^{2} \) |
| 17 | \( 1 + (-1.69 - 0.125i)T + (0.989 + 0.146i)T^{2} \) |
| 23 | \( 1 + (0.703 + 0.710i)T^{2} \) |
| 29 | \( 1 + (-0.663 - 0.748i)T^{2} \) |
| 31 | \( 1 + (0.962 - 0.272i)T^{2} \) |
| 37 | \( 1 + (0.986 - 0.164i)T^{2} \) |
| 41 | \( 1 + (-1.65 + 0.638i)T + (0.741 - 0.670i)T^{2} \) |
| 43 | \( 1 + (1.82 + 0.338i)T + (0.933 + 0.359i)T^{2} \) |
| 47 | \( 1 + (-0.983 - 0.182i)T^{2} \) |
| 53 | \( 1 + (0.649 - 0.760i)T^{2} \) |
| 59 | \( 1 + (0.0652 - 0.414i)T + (-0.951 - 0.307i)T^{2} \) |
| 61 | \( 1 + (0.0459 - 0.998i)T^{2} \) |
| 67 | \( 1 + (0.438 + 0.292i)T + (0.384 + 0.922i)T^{2} \) |
| 71 | \( 1 + (0.0459 + 0.998i)T^{2} \) |
| 73 | \( 1 + (0.812 + 1.68i)T + (-0.621 + 0.783i)T^{2} \) |
| 79 | \( 1 + (0.155 + 0.987i)T^{2} \) |
| 83 | \( 1 + (-1.13 + 0.695i)T + (0.451 - 0.892i)T^{2} \) |
| 89 | \( 1 + (0.0239 - 0.0496i)T + (-0.621 - 0.783i)T^{2} \) |
| 97 | \( 1 + (1.56 - 1.04i)T + (0.384 - 0.922i)T^{2} \) |
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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
−8.957887145906649256755824999929, −8.280158378020528164104090810052, −7.63904472657648021485289698253, −7.44224934296574332297203897455, −6.22980518290131651833993286603, −5.25033533809855206041946867061, −4.50153666182831789577511395854, −3.53848896631938903417531157735, −2.56380077021705471343085877685, −1.56263104961408482035305585074,
1.09654179214594970666624645779, 2.32031487130582215852068179409, 3.01910370878262495353415145458, 3.41870186364703784179258661127, 4.56481592745711151621686534909, 5.50776668993689776242674025068, 6.96851788458510298123306065272, 7.67707612904437007814008998183, 8.282882486985135374998043700796, 8.615554998612859315516206733793