L(s) = 1 | + (−1.31 − 3.18i)3-s + (0.659 − 1.59i)5-s + (−9.54 + 9.54i)7-s + (−2.03 + 2.03i)9-s + (−3.96 + 9.57i)11-s + (−1.91 − 4.63i)13-s − 5.93·15-s + 15.3i·17-s + (0.827 − 0.342i)19-s + (42.9 + 17.8i)21-s + (12.9 + 12.9i)23-s + (15.5 + 15.5i)25-s + (−19.5 − 8.07i)27-s + (−23.7 + 9.85i)29-s − 25.1i·31-s + ⋯ |
L(s) = 1 | + (−0.439 − 1.06i)3-s + (0.131 − 0.318i)5-s + (−1.36 + 1.36i)7-s + (−0.225 + 0.225i)9-s + (−0.360 + 0.870i)11-s + (−0.147 − 0.356i)13-s − 0.395·15-s + 0.900i·17-s + (0.0435 − 0.0180i)19-s + (2.04 + 0.847i)21-s + (0.561 + 0.561i)23-s + (0.623 + 0.623i)25-s + (−0.722 − 0.299i)27-s + (−0.820 + 0.339i)29-s − 0.811i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0818 - 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0818 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.346198 + 0.375786i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.346198 + 0.375786i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (1.31 + 3.18i)T + (-6.36 + 6.36i)T^{2} \) |
| 5 | \( 1 + (-0.659 + 1.59i)T + (-17.6 - 17.6i)T^{2} \) |
| 7 | \( 1 + (9.54 - 9.54i)T - 49iT^{2} \) |
| 11 | \( 1 + (3.96 - 9.57i)T + (-85.5 - 85.5i)T^{2} \) |
| 13 | \( 1 + (1.91 + 4.63i)T + (-119. + 119. i)T^{2} \) |
| 17 | \( 1 - 15.3iT - 289T^{2} \) |
| 19 | \( 1 + (-0.827 + 0.342i)T + (255. - 255. i)T^{2} \) |
| 23 | \( 1 + (-12.9 - 12.9i)T + 529iT^{2} \) |
| 29 | \( 1 + (23.7 - 9.85i)T + (594. - 594. i)T^{2} \) |
| 31 | \( 1 + 25.1iT - 961T^{2} \) |
| 37 | \( 1 + (13.6 - 32.8i)T + (-968. - 968. i)T^{2} \) |
| 41 | \( 1 + (32.9 - 32.9i)T - 1.68e3iT^{2} \) |
| 43 | \( 1 + (17.9 - 43.3i)T + (-1.30e3 - 1.30e3i)T^{2} \) |
| 47 | \( 1 + 20.1T + 2.20e3T^{2} \) |
| 53 | \( 1 + (35.0 + 14.5i)T + (1.98e3 + 1.98e3i)T^{2} \) |
| 59 | \( 1 + (60.6 + 25.1i)T + (2.46e3 + 2.46e3i)T^{2} \) |
| 61 | \( 1 + (-27.9 + 11.5i)T + (2.63e3 - 2.63e3i)T^{2} \) |
| 67 | \( 1 + (-1.13 - 2.73i)T + (-3.17e3 + 3.17e3i)T^{2} \) |
| 71 | \( 1 + (-45.6 + 45.6i)T - 5.04e3iT^{2} \) |
| 73 | \( 1 + (29.1 - 29.1i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + 3.27T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-56.7 + 23.5i)T + (4.87e3 - 4.87e3i)T^{2} \) |
| 89 | \( 1 + (44.5 + 44.5i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + 106.T + 9.40e3T^{2} \) |
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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
−12.34047099835866879691744068955, −11.41154875169815160328018744217, −9.928710627600385076537435713532, −9.254873362172771557073669616510, −8.040820153314029190248321560565, −6.87888852646937075556980283036, −6.12294073516844712128294379628, −5.16939943710715307021906208564, −3.17525543757390487955127689391, −1.75662149613657879672877791118,
0.26978632066311563247783383332, 3.10421318203806112189755667025, 4.05395424933698099078255558934, 5.24528270042529075000295913873, 6.53808770741529740743309075619, 7.33659936891790847083551157903, 8.974255570802694599046895490449, 9.892016295671650294534824104488, 10.53932534614186787389334225707, 11.11033577282273985738222760070