L(s) = 1 | − 4i·7-s − 4·11-s − 6i·13-s − 2i·17-s − 4·19-s + 10·29-s − 4·31-s − 10i·37-s − 2·41-s + 4i·43-s − 8i·47-s − 9·49-s + 2i·53-s + 12·59-s − 10·61-s + ⋯ |
L(s) = 1 | − 1.51i·7-s − 1.20·11-s − 1.66i·13-s − 0.485i·17-s − 0.917·19-s + 1.85·29-s − 0.718·31-s − 1.64i·37-s − 0.312·41-s + 0.609i·43-s − 1.16i·47-s − 1.28·49-s + 0.274i·53-s + 1.56·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7646040205\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7646040205\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 10T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 14iT - 97T^{2} \) |
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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
−7.47506155042977246152706259621, −7.05835982154405024506642021281, −6.10592080573278282868956761245, −5.33581199558648462467141079639, −4.70338864632780707424900770710, −3.88661337138591488107247154971, −3.09842722811340032224443159509, −2.32943717001284080000827350318, −0.930206401012408623769894772979, −0.20892009420268735719382647303,
1.61055313714174325201450030821, 2.36345801606063690494396420450, 2.96851023374734845740381120084, 4.14097509146134042914261592123, 4.86181447517141696551655948852, 5.45652324255686854982279037924, 6.40404807488187302707456452005, 6.63142690775718091741646516440, 7.80049672024111089752290395201, 8.427469013565420073431181341357