| L(s) = 1 | − 3-s − 2·5-s − 4·7-s + 9-s + 11-s + 6·13-s + 2·15-s − 2·17-s + 4·19-s + 4·21-s + 4·23-s − 25-s − 27-s − 2·29-s − 8·31-s − 33-s + 8·35-s − 37-s − 6·39-s + 6·41-s + 4·43-s − 2·45-s + 8·47-s + 9·49-s + 2·51-s + 2·53-s − 2·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 1.51·7-s + 1/3·9-s + 0.301·11-s + 1.66·13-s + 0.516·15-s − 0.485·17-s + 0.917·19-s + 0.872·21-s + 0.834·23-s − 1/5·25-s − 0.192·27-s − 0.371·29-s − 1.43·31-s − 0.174·33-s + 1.35·35-s − 0.164·37-s − 0.960·39-s + 0.937·41-s + 0.609·43-s − 0.298·45-s + 1.16·47-s + 9/7·49-s + 0.280·51-s + 0.274·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.367333975\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.367333975\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| 37 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.90395679908272, −13.42170578669694, −12.85583398939576, −12.66524024735478, −12.03883821577103, −11.41075752038295, −11.05936930774153, −10.78527266401177, −9.926209108308311, −9.523352902483125, −8.946508720745982, −8.612356350006179, −7.759545284606629, −7.248327493475333, −6.833687804431496, −6.286811486418506, −5.710270780735303, −5.375763907968522, −4.326992665047472, −3.854392624783888, −3.531649374223917, −2.919593505049284, −1.970650116962920, −0.9667913686112918, −0.5033710534316353,
0.5033710534316353, 0.9667913686112918, 1.970650116962920, 2.919593505049284, 3.531649374223917, 3.854392624783888, 4.326992665047472, 5.375763907968522, 5.710270780735303, 6.286811486418506, 6.833687804431496, 7.248327493475333, 7.759545284606629, 8.612356350006179, 8.946508720745982, 9.523352902483125, 9.926209108308311, 10.78527266401177, 11.05936930774153, 11.41075752038295, 12.03883821577103, 12.66524024735478, 12.85583398939576, 13.42170578669694, 13.90395679908272
