L(s) = 1 | + (0.309 + 0.951i)2-s + (0.355 + 1.69i)3-s + (−0.809 + 0.587i)4-s + (−2.40 − 0.780i)5-s + (−1.50 + 0.861i)6-s + (−0.587 − 0.809i)7-s + (−0.809 − 0.587i)8-s + (−2.74 + 1.20i)9-s − 2.52i·10-s + (−2.28 − 2.40i)11-s + (−1.28 − 1.16i)12-s + (−2.68 + 0.871i)13-s + (0.587 − 0.809i)14-s + (0.469 − 4.34i)15-s + (0.309 − 0.951i)16-s + (−0.600 + 1.84i)17-s + ⋯ |
L(s) = 1 | + (0.218 + 0.672i)2-s + (0.205 + 0.978i)3-s + (−0.404 + 0.293i)4-s + (−1.07 − 0.348i)5-s + (−0.613 + 0.351i)6-s + (−0.222 − 0.305i)7-s + (−0.286 − 0.207i)8-s + (−0.915 + 0.401i)9-s − 0.798i·10-s + (−0.689 − 0.724i)11-s + (−0.370 − 0.335i)12-s + (−0.743 + 0.241i)13-s + (0.157 − 0.216i)14-s + (0.121 − 1.12i)15-s + (0.0772 − 0.237i)16-s + (−0.145 + 0.448i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.549 + 0.835i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.549 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.138614 - 0.257070i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.138614 - 0.257070i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.309 - 0.951i)T \) |
| 3 | \( 1 + (-0.355 - 1.69i)T \) |
| 7 | \( 1 + (0.587 + 0.809i)T \) |
| 11 | \( 1 + (2.28 + 2.40i)T \) |
good | 5 | \( 1 + (2.40 + 0.780i)T + (4.04 + 2.93i)T^{2} \) |
| 13 | \( 1 + (2.68 - 0.871i)T + (10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (0.600 - 1.84i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (0.670 - 0.922i)T + (-5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 - 1.95iT - 23T^{2} \) |
| 29 | \( 1 + (0.436 - 0.316i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-1.43 - 4.41i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (4.42 - 3.21i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-4.71 - 3.42i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 7.17iT - 43T^{2} \) |
| 47 | \( 1 + (6.77 - 9.32i)T + (-14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-10.7 + 3.49i)T + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-2.15 - 2.96i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (9.83 + 3.19i)T + (49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + 0.0777T + 67T^{2} \) |
| 71 | \( 1 + (11.1 + 3.62i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-4.64 - 6.39i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (16.0 - 5.22i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (0.819 - 2.52i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + 0.938iT - 89T^{2} \) |
| 97 | \( 1 + (-0.0157 - 0.0484i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−11.59840931028620230813362281132, −10.67698655672726776801939338226, −9.795240603945129722788083602054, −8.695160451849961959440037326011, −8.127816895553374322970726528245, −7.20719889058300964289959474649, −5.84800193106604829173294724561, −4.83919389304740628149612569276, −4.03256560406733810495458313008, −3.06042301867208128647368536790,
0.15469517869626656293239736883, 2.22592813334209483017070343086, 3.11137618844591019427604879138, 4.43946884402370089483790142957, 5.66425040197278826748157724953, 7.00807035070085676314522030670, 7.62218816454085071851387084491, 8.561941611646876956237546075564, 9.634079085309426963868551816180, 10.67783175160082305208848269535