| L(s) = 1 | − 3-s − 2·7-s + 9-s + 2·13-s − 17-s − 4·19-s + 2·21-s − 2·23-s − 5·25-s − 27-s − 6·31-s − 2·39-s − 10·41-s − 4·43-s + 4·47-s − 3·49-s + 51-s − 2·53-s + 4·57-s + 4·59-s − 2·63-s − 4·67-s + 2·69-s + 2·71-s − 14·73-s + 5·75-s − 6·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s + 0.554·13-s − 0.242·17-s − 0.917·19-s + 0.436·21-s − 0.417·23-s − 25-s − 0.192·27-s − 1.07·31-s − 0.320·39-s − 1.56·41-s − 0.609·43-s + 0.583·47-s − 3/7·49-s + 0.140·51-s − 0.274·53-s + 0.529·57-s + 0.520·59-s − 0.251·63-s − 0.488·67-s + 0.240·69-s + 0.237·71-s − 1.63·73-s + 0.577·75-s − 0.675·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 17 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−9.941611251678217120793425269173, −9.041019508128166537998788535519, −8.139884873932014733298462713743, −7.03118715673001635895164549850, −6.29392484047154187578719849324, −5.53399853944419593600376639581, −4.32146836767987654020354473360, −3.38688660560456486653785466782, −1.86322179819912387769835446845, 0,
1.86322179819912387769835446845, 3.38688660560456486653785466782, 4.32146836767987654020354473360, 5.53399853944419593600376639581, 6.29392484047154187578719849324, 7.03118715673001635895164549850, 8.139884873932014733298462713743, 9.041019508128166537998788535519, 9.941611251678217120793425269173
