L(s) = 1 | + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)9-s + 1.73i·13-s + (0.5 − 0.866i)19-s + (0.5 + 0.866i)25-s − 0.999·27-s + (−0.5 − 0.866i)31-s + (−0.5 + 0.866i)37-s + (−1.49 + 0.866i)39-s + 1.73i·43-s + 0.999·57-s + (1.5 − 0.866i)67-s + (1.5 − 0.866i)73-s + (−0.499 + 0.866i)75-s + (−1.5 − 0.866i)79-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)9-s + 1.73i·13-s + (0.5 − 0.866i)19-s + (0.5 + 0.866i)25-s − 0.999·27-s + (−0.5 − 0.866i)31-s + (−0.5 + 0.866i)37-s + (−1.49 + 0.866i)39-s + 1.73i·43-s + 0.999·57-s + (1.5 − 0.866i)67-s + (1.5 − 0.866i)73-s + (−0.499 + 0.866i)75-s + (−1.5 − 0.866i)79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.329148848\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.329148848\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 - 1.73iT - T^{2} \) |
| 17 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - 1.73iT - T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.386725719406232762259228325723, −8.829431542911691174532891929046, −7.943640985036925301273516970132, −7.10305047058641789067305016253, −6.30792394344717580039821369900, −5.16571731499131676393689109910, −4.57264677772177362081331077248, −3.72928945178457173015163671959, −2.81490086691775600320431139664, −1.74336759877131082053515560156,
0.876804002733165351353712806347, 2.15346985224634997342782038706, 3.11110430347519644808400840119, 3.82989702312932093486584324411, 5.29843358412518647895401140583, 5.79350615601559685751274934500, 6.85300142354819138364791964514, 7.44131565601255858007011060816, 8.285130916565853550874192986572, 8.632883593553366184180235509258