| L(s) = 1 | − 3-s + 4·5-s − 2·7-s + 9-s − 11-s − 4·15-s + 2·17-s + 4·19-s + 2·21-s − 6·23-s + 11·25-s − 27-s − 6·29-s + 4·31-s + 33-s − 8·35-s + 2·37-s − 6·41-s − 8·43-s + 4·45-s + 10·47-s − 3·49-s − 2·51-s − 12·53-s − 4·55-s − 4·57-s + 8·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.78·5-s − 0.755·7-s + 1/3·9-s − 0.301·11-s − 1.03·15-s + 0.485·17-s + 0.917·19-s + 0.436·21-s − 1.25·23-s + 11/5·25-s − 0.192·27-s − 1.11·29-s + 0.718·31-s + 0.174·33-s − 1.35·35-s + 0.328·37-s − 0.937·41-s − 1.21·43-s + 0.596·45-s + 1.45·47-s − 3/7·49-s − 0.280·51-s − 1.64·53-s − 0.539·55-s − 0.529·57-s + 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 178464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 178464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−13.27109966076374, −13.07706127009640, −12.57267996473690, −11.99795088696488, −11.58509508264352, −10.99101160410556, −10.29189403476998, −10.12593901223924, −9.689611566400095, −9.402137895319775, −8.767463488441193, −8.146037065971711, −7.509366665601754, −7.026256235260846, −6.321060341082433, −6.145959326178263, −5.697643249746203, −5.080421138367335, −4.858235416786414, −3.834927067544348, −3.352481964968773, −2.699925535363522, −2.079420636079522, −1.576261235497646, −0.8801378320701893, 0,
0.8801378320701893, 1.576261235497646, 2.079420636079522, 2.699925535363522, 3.352481964968773, 3.834927067544348, 4.858235416786414, 5.080421138367335, 5.697643249746203, 6.145959326178263, 6.321060341082433, 7.026256235260846, 7.509366665601754, 8.146037065971711, 8.767463488441193, 9.402137895319775, 9.689611566400095, 10.12593901223924, 10.29189403476998, 10.99101160410556, 11.58509508264352, 11.99795088696488, 12.57267996473690, 13.07706127009640, 13.27109966076374
