L(s) = 1 | − 6.06i·5-s − 2.64·7-s − 12.1i·11-s + 18.5·13-s + 10.9i·17-s − 20·19-s + 12.1i·23-s − 11.8·25-s + 41.8i·29-s + 25.1·31-s + 16.0i·35-s − 38·37-s + 60.6i·41-s − 83.4·43-s + 16.9i·47-s + ⋯ |
L(s) = 1 | − 1.21i·5-s − 0.377·7-s − 1.10i·11-s + 1.42·13-s + 0.641i·17-s − 1.05·19-s + 0.527i·23-s − 0.473·25-s + 1.44i·29-s + 0.811·31-s + 0.458i·35-s − 1.02·37-s + 1.47i·41-s − 1.94·43-s + 0.361i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.268618249\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.268618249\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 2.64T \) |
good | 5 | \( 1 + 6.06iT - 25T^{2} \) |
| 11 | \( 1 + 12.1iT - 121T^{2} \) |
| 13 | \( 1 - 18.5T + 169T^{2} \) |
| 17 | \( 1 - 10.9iT - 289T^{2} \) |
| 19 | \( 1 + 20T + 361T^{2} \) |
| 23 | \( 1 - 12.1iT - 529T^{2} \) |
| 29 | \( 1 - 41.8iT - 841T^{2} \) |
| 31 | \( 1 - 25.1T + 961T^{2} \) |
| 37 | \( 1 + 38T + 1.36e3T^{2} \) |
| 41 | \( 1 - 60.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 83.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 16.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 94.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 58.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 15.6T + 3.72e3T^{2} \) |
| 67 | \( 1 - 132.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 12.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 76.9T + 5.32e3T^{2} \) |
| 79 | \( 1 - 33.6T + 6.24e3T^{2} \) |
| 83 | \( 1 + 60.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 4.77iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 188.T + 9.40e3T^{2} \) |
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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
−8.592327363919891475166855192046, −7.985293907462415471774184372007, −6.67119606273900714210942195015, −6.19713413776298766594326229807, −5.42458489126571379834774246313, −4.65113826151124622572342536638, −3.73460274532662397689047482937, −3.12503203666539465076982390309, −1.61200068733452187481589994226, −0.977512718964534596528642118833,
0.29335694448670033507844385990, 1.84038682333968100088694703488, 2.59622242783980235021720706738, 3.53452980492984024759592298427, 4.18587943126388283080875211853, 5.20385039673319387007013983499, 6.23238778085137755214282914728, 6.70467427237114503817602132474, 7.17907930852523395144448293707, 8.242947967176902022001033956349