| L(s) = 1 | − 66·5-s + 176·7-s − 60·11-s + 658·13-s + 414·17-s − 956·19-s − 600·23-s + 1.23e3·25-s + 5.57e3·29-s − 3.59e3·31-s − 1.16e4·35-s + 8.45e3·37-s − 1.91e4·41-s − 1.33e4·43-s + 1.96e4·47-s + 1.41e4·49-s − 3.12e4·53-s + 3.96e3·55-s + 2.63e4·59-s + 3.10e4·61-s − 4.34e4·65-s + 1.68e4·67-s − 6.12e3·71-s − 2.55e4·73-s − 1.05e4·77-s + 7.44e4·79-s − 6.46e3·83-s + ⋯ |
| L(s) = 1 | − 1.18·5-s + 1.35·7-s − 0.149·11-s + 1.07·13-s + 0.347·17-s − 0.607·19-s − 0.236·23-s + 0.393·25-s + 1.23·29-s − 0.671·31-s − 1.60·35-s + 1.01·37-s − 1.78·41-s − 1.09·43-s + 1.29·47-s + 0.843·49-s − 1.52·53-s + 0.176·55-s + 0.985·59-s + 1.06·61-s − 1.27·65-s + 0.457·67-s − 0.144·71-s − 0.561·73-s − 0.202·77-s + 1.34·79-s − 0.103·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.993691158\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.993691158\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 66 T + p^{5} T^{2} \) |
| 7 | \( 1 - 176 T + p^{5} T^{2} \) |
| 11 | \( 1 + 60 T + p^{5} T^{2} \) |
| 13 | \( 1 - 658 T + p^{5} T^{2} \) |
| 17 | \( 1 - 414 T + p^{5} T^{2} \) |
| 19 | \( 1 + 956 T + p^{5} T^{2} \) |
| 23 | \( 1 + 600 T + p^{5} T^{2} \) |
| 29 | \( 1 - 5574 T + p^{5} T^{2} \) |
| 31 | \( 1 + 3592 T + p^{5} T^{2} \) |
| 37 | \( 1 - 8458 T + p^{5} T^{2} \) |
| 41 | \( 1 + 19194 T + p^{5} T^{2} \) |
| 43 | \( 1 + 13316 T + p^{5} T^{2} \) |
| 47 | \( 1 - 19680 T + p^{5} T^{2} \) |
| 53 | \( 1 + 31266 T + p^{5} T^{2} \) |
| 59 | \( 1 - 26340 T + p^{5} T^{2} \) |
| 61 | \( 1 - 31090 T + p^{5} T^{2} \) |
| 67 | \( 1 - 16804 T + p^{5} T^{2} \) |
| 71 | \( 1 + 6120 T + p^{5} T^{2} \) |
| 73 | \( 1 + 25558 T + p^{5} T^{2} \) |
| 79 | \( 1 - 74408 T + p^{5} T^{2} \) |
| 83 | \( 1 + 6468 T + p^{5} T^{2} \) |
| 89 | \( 1 - 32742 T + p^{5} T^{2} \) |
| 97 | \( 1 - 166082 T + p^{5} T^{2} \) |
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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
−10.12401073648237253639057232665, −8.645170081650686857178235100828, −8.248417550144528325919526659679, −7.50297091186964700006399535038, −6.37533545020173563394973695377, −5.14092382956746453208076139859, −4.29005087198710159935171610572, −3.40285054047205746393702916554, −1.87178861134642040974138842596, −0.71081050348727881836608922931,
0.71081050348727881836608922931, 1.87178861134642040974138842596, 3.40285054047205746393702916554, 4.29005087198710159935171610572, 5.14092382956746453208076139859, 6.37533545020173563394973695377, 7.50297091186964700006399535038, 8.248417550144528325919526659679, 8.645170081650686857178235100828, 10.12401073648237253639057232665
