| L(s) = 1 | + (1 + 2i)5-s − 2i·7-s + 11-s − 4i·17-s − 4·19-s + 6i·23-s + (−3 + 4i)25-s + 2·29-s + 8·31-s + (4 − 2i)35-s + 4i·37-s + 6·41-s − 6i·43-s − 2i·47-s + 3·49-s + ⋯ |
| L(s) = 1 | + (0.447 + 0.894i)5-s − 0.755i·7-s + 0.301·11-s − 0.970i·17-s − 0.917·19-s + 1.25i·23-s + (−0.600 + 0.800i)25-s + 0.371·29-s + 1.43·31-s + (0.676 − 0.338i)35-s + 0.657i·37-s + 0.937·41-s − 0.914i·43-s − 0.291i·47-s + 0.428·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.064612533\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.064612533\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1 - 2i)T \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 + 2iT - 47T^{2} \) |
| 53 | \( 1 - 12iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 14T + 61T^{2} \) |
| 67 | \( 1 + 10iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 2iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 8iT - 97T^{2} \) |
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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
−8.509151204991037720525387181518, −7.56380120870672226378041199914, −7.07471794438814770083035953668, −6.39804456224879840857164182770, −5.66882303551193056114112541739, −4.67497077699503002441332596121, −3.86762120371455746166277772484, −3.00571192406293325630737643907, −2.15097134920176642783220814059, −0.913959038410848158142972276161,
0.76682447496435372864109877362, 1.95780733126578720706026041785, 2.65791307822691878667918729706, 4.02440603694632319159604924134, 4.58221927067700631649930411133, 5.48925641320362647852716259597, 6.17149480136203216358665343108, 6.68899086485885057007120434305, 7.969278943331290246984175117134, 8.561690076916054853886704956183
