| L(s) = 1 | − 4·2-s − 9·3-s + 16·4-s − 14·5-s + 36·6-s + 130·7-s − 64·8-s + 81·9-s + 56·10-s − 121·11-s − 144·12-s − 122·13-s − 520·14-s + 126·15-s + 256·16-s − 1.10e3·17-s − 324·18-s − 1.31e3·19-s − 224·20-s − 1.17e3·21-s + 484·22-s − 3.00e3·23-s + 576·24-s − 2.92e3·25-s + 488·26-s − 729·27-s + 2.08e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.250·5-s + 0.408·6-s + 1.00·7-s − 0.353·8-s + 1/3·9-s + 0.177·10-s − 0.301·11-s − 0.288·12-s − 0.200·13-s − 0.709·14-s + 0.144·15-s + 1/4·16-s − 0.929·17-s − 0.235·18-s − 0.835·19-s − 0.125·20-s − 0.578·21-s + 0.213·22-s − 1.18·23-s + 0.204·24-s − 0.937·25-s + 0.141·26-s − 0.192·27-s + 0.501·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p^{2} T \) |
| 11 | \( 1 + p^{2} T \) |
| good | 5 | \( 1 + 14 T + p^{5} T^{2} \) |
| 7 | \( 1 - 130 T + p^{5} T^{2} \) |
| 13 | \( 1 + 122 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1108 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1314 T + p^{5} T^{2} \) |
| 23 | \( 1 + 3000 T + p^{5} T^{2} \) |
| 29 | \( 1 + 4432 T + p^{5} T^{2} \) |
| 31 | \( 1 - 880 T + p^{5} T^{2} \) |
| 37 | \( 1 - 11818 T + p^{5} T^{2} \) |
| 41 | \( 1 + 5648 T + p^{5} T^{2} \) |
| 43 | \( 1 - 778 T + p^{5} T^{2} \) |
| 47 | \( 1 + 10672 T + p^{5} T^{2} \) |
| 53 | \( 1 - 9086 T + p^{5} T^{2} \) |
| 59 | \( 1 + 12012 T + p^{5} T^{2} \) |
| 61 | \( 1 - 39826 T + p^{5} T^{2} \) |
| 67 | \( 1 + 56316 T + p^{5} T^{2} \) |
| 71 | \( 1 + 51920 T + p^{5} T^{2} \) |
| 73 | \( 1 + 10266 T + p^{5} T^{2} \) |
| 79 | \( 1 + 79646 T + p^{5} T^{2} \) |
| 83 | \( 1 - 30224 T + p^{5} T^{2} \) |
| 89 | \( 1 + 75310 T + p^{5} T^{2} \) |
| 97 | \( 1 + 43778 T + p^{5} T^{2} \) |
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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
−13.15141053579592860071343945647, −11.77955559266127482812358503346, −11.09570059637665710218651495279, −9.958498739622676393797525978312, −8.486317612316221000289019445398, −7.47718841639675707002042532776, −5.99089763546447393190320708379, −4.40359096155344720222921178447, −1.94049119232238304225214754783, 0,
1.94049119232238304225214754783, 4.40359096155344720222921178447, 5.99089763546447393190320708379, 7.47718841639675707002042532776, 8.486317612316221000289019445398, 9.958498739622676393797525978312, 11.09570059637665710218651495279, 11.77955559266127482812358503346, 13.15141053579592860071343945647
