| L(s) = 1 | + 10·2-s + 68·4-s − 56·5-s + 360·8-s − 560·10-s − 232·11-s + 140·13-s + 1.42e3·16-s − 1.72e3·17-s + 98·19-s − 3.80e3·20-s − 2.32e3·22-s − 1.82e3·23-s + 11·25-s + 1.40e3·26-s − 3.41e3·29-s + 7.64e3·31-s + 2.72e3·32-s − 1.72e4·34-s − 1.03e4·37-s + 980·38-s − 2.01e4·40-s − 1.79e4·41-s + 1.08e4·43-s − 1.57e4·44-s − 1.82e4·46-s + 9.32e3·47-s + ⋯ |
| L(s) = 1 | + 1.76·2-s + 17/8·4-s − 1.00·5-s + 1.98·8-s − 1.77·10-s − 0.578·11-s + 0.229·13-s + 1.39·16-s − 1.44·17-s + 0.0622·19-s − 2.12·20-s − 1.02·22-s − 0.718·23-s + 0.00351·25-s + 0.406·26-s − 0.754·29-s + 1.42·31-s + 0.469·32-s − 2.55·34-s − 1.24·37-s + 0.110·38-s − 1.99·40-s − 1.66·41-s + 0.897·43-s − 1.22·44-s − 1.27·46-s + 0.615·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 5 p T + p^{5} T^{2} \) |
| 5 | \( 1 + 56 T + p^{5} T^{2} \) |
| 11 | \( 1 + 232 T + p^{5} T^{2} \) |
| 13 | \( 1 - 140 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1722 T + p^{5} T^{2} \) |
| 19 | \( 1 - 98 T + p^{5} T^{2} \) |
| 23 | \( 1 + 1824 T + p^{5} T^{2} \) |
| 29 | \( 1 + 3418 T + p^{5} T^{2} \) |
| 31 | \( 1 - 7644 T + p^{5} T^{2} \) |
| 37 | \( 1 + 10398 T + p^{5} T^{2} \) |
| 41 | \( 1 + 17962 T + p^{5} T^{2} \) |
| 43 | \( 1 - 10880 T + p^{5} T^{2} \) |
| 47 | \( 1 - 9324 T + p^{5} T^{2} \) |
| 53 | \( 1 + 2262 T + p^{5} T^{2} \) |
| 59 | \( 1 + 2730 T + p^{5} T^{2} \) |
| 61 | \( 1 + 25648 T + p^{5} T^{2} \) |
| 67 | \( 1 + 48404 T + p^{5} T^{2} \) |
| 71 | \( 1 - 58560 T + p^{5} T^{2} \) |
| 73 | \( 1 + 68082 T + p^{5} T^{2} \) |
| 79 | \( 1 - 31784 T + p^{5} T^{2} \) |
| 83 | \( 1 + 20538 T + p^{5} T^{2} \) |
| 89 | \( 1 + 50582 T + p^{5} T^{2} \) |
| 97 | \( 1 - 58506 T + p^{5} T^{2} \) |
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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
−10.33145570929293011485337795167, −8.734901348613176033973460375295, −7.69420423533473828724706572863, −6.81957346608229485910790327923, −5.88572278957813546691275192521, −4.78461159818991410679145984901, −4.09144905346805858433115117504, −3.16081894666775625652676117077, −2.01625118911369290174183254413, 0,
2.01625118911369290174183254413, 3.16081894666775625652676117077, 4.09144905346805858433115117504, 4.78461159818991410679145984901, 5.88572278957813546691275192521, 6.81957346608229485910790327923, 7.69420423533473828724706572863, 8.734901348613176033973460375295, 10.33145570929293011485337795167
