| L(s) = 1 | − 4·2-s + 9·3-s + 16·4-s − 54·5-s − 36·6-s + 104·7-s − 64·8-s + 81·9-s + 216·10-s − 330·11-s + 144·12-s − 46·13-s − 416·14-s − 486·15-s + 256·16-s − 618·17-s − 324·18-s + 361·19-s − 864·20-s + 936·21-s + 1.32e3·22-s − 402·23-s − 576·24-s − 209·25-s + 184·26-s + 729·27-s + 1.66e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.965·5-s − 0.408·6-s + 0.802·7-s − 0.353·8-s + 1/3·9-s + 0.683·10-s − 0.822·11-s + 0.288·12-s − 0.0754·13-s − 0.567·14-s − 0.557·15-s + 1/4·16-s − 0.518·17-s − 0.235·18-s + 0.229·19-s − 0.482·20-s + 0.463·21-s + 0.581·22-s − 0.158·23-s − 0.204·24-s − 0.0668·25-s + 0.0533·26-s + 0.192·27-s + 0.401·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + p^{2} T \) |
| 3 | \( 1 - p^{2} T \) |
| 19 | \( 1 - p^{2} T \) |
| good | 5 | \( 1 + 54 T + p^{5} T^{2} \) |
| 7 | \( 1 - 104 T + p^{5} T^{2} \) |
| 11 | \( 1 + 30 p T + p^{5} T^{2} \) |
| 13 | \( 1 + 46 T + p^{5} T^{2} \) |
| 17 | \( 1 + 618 T + p^{5} T^{2} \) |
| 23 | \( 1 + 402 T + p^{5} T^{2} \) |
| 29 | \( 1 + 2628 T + p^{5} T^{2} \) |
| 31 | \( 1 + 2368 T + p^{5} T^{2} \) |
| 37 | \( 1 + 12130 T + p^{5} T^{2} \) |
| 41 | \( 1 + 18864 T + p^{5} T^{2} \) |
| 43 | \( 1 + 10408 T + p^{5} T^{2} \) |
| 47 | \( 1 + 4770 T + p^{5} T^{2} \) |
| 53 | \( 1 + 19452 T + p^{5} T^{2} \) |
| 59 | \( 1 - 30528 T + p^{5} T^{2} \) |
| 61 | \( 1 - 11138 T + p^{5} T^{2} \) |
| 67 | \( 1 - 49508 T + p^{5} T^{2} \) |
| 71 | \( 1 - 7572 T + p^{5} T^{2} \) |
| 73 | \( 1 - 2342 T + p^{5} T^{2} \) |
| 79 | \( 1 - 22424 T + p^{5} T^{2} \) |
| 83 | \( 1 + 46734 T + p^{5} T^{2} \) |
| 89 | \( 1 + 70104 T + p^{5} T^{2} \) |
| 97 | \( 1 - 105710 T + p^{5} T^{2} \) |
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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
−11.87602005177780782341279750545, −11.05478115143156214223463318381, −9.949878674307051229738921057254, −8.583047251985726636876231932724, −7.974525530368315065036718200213, −7.01047732071316283938506524179, −5.05565544974233652585307204564, −3.51700168806837292559859712960, −1.89961094823661015590398451946, 0,
1.89961094823661015590398451946, 3.51700168806837292559859712960, 5.05565544974233652585307204564, 7.01047732071316283938506524179, 7.974525530368315065036718200213, 8.583047251985726636876231932724, 9.949878674307051229738921057254, 11.05478115143156214223463318381, 11.87602005177780782341279750545
