L(s) = 1 | + 2.44i·5-s − 3.46·7-s + 8.48i·11-s + 10.3·13-s + 12.7i·17-s + 4·19-s + 34.2i·23-s + 19·25-s − 46.5i·29-s + 38.1·31-s − 8.48i·35-s − 27.7·37-s − 29.6i·41-s − 56·43-s − 63.6i·47-s + ⋯ |
L(s) = 1 | + 0.489i·5-s − 0.494·7-s + 0.771i·11-s + 0.799·13-s + 0.748i·17-s + 0.210·19-s + 1.49i·23-s + 0.760·25-s − 1.60i·29-s + 1.22·31-s − 0.242i·35-s − 0.748·37-s − 0.724i·41-s − 1.30·43-s − 1.35i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.425636671\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.425636671\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 2.44iT - 25T^{2} \) |
| 7 | \( 1 + 3.46T + 49T^{2} \) |
| 11 | \( 1 - 8.48iT - 121T^{2} \) |
| 13 | \( 1 - 10.3T + 169T^{2} \) |
| 17 | \( 1 - 12.7iT - 289T^{2} \) |
| 19 | \( 1 - 4T + 361T^{2} \) |
| 23 | \( 1 - 34.2iT - 529T^{2} \) |
| 29 | \( 1 + 46.5iT - 841T^{2} \) |
| 31 | \( 1 - 38.1T + 961T^{2} \) |
| 37 | \( 1 + 27.7T + 1.36e3T^{2} \) |
| 41 | \( 1 + 29.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 56T + 1.84e3T^{2} \) |
| 47 | \( 1 + 63.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 71.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 101. iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 69.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + 28T + 4.48e3T^{2} \) |
| 71 | \( 1 - 53.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 20T + 5.32e3T^{2} \) |
| 79 | \( 1 - 72.7T + 6.24e3T^{2} \) |
| 83 | \( 1 - 76.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 55.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 8T + 9.40e3T^{2} \) |
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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
−9.089862304916994463022465350851, −8.329776090819620502492068104077, −7.48787663153848902998637632098, −6.74472263460840411687038198944, −6.08938886927974881332495745903, −5.21391203758571193131283406822, −4.10034167787061942483635509115, −3.41044219146073083077993268584, −2.37352115358092387477042796130, −1.23671756793364779417381779923,
0.37618521451779055772731825940, 1.37387186671812639405674641663, 2.85790340954569303426290853647, 3.47550252188685458212256777944, 4.69667679687453910184354155086, 5.25849218435432718568825888695, 6.49021607155980130429387864909, 6.67398573256256372693711852992, 8.045962493489796647751130303242, 8.547795650329847115812861620463