| L(s) = 1 | + 2·3-s − 2·5-s − 4·7-s + 9-s − 2·11-s − 2·13-s − 4·15-s + 2·17-s + 2·19-s − 8·21-s + 4·23-s − 25-s − 4·27-s − 4·33-s + 8·35-s + 10·37-s − 4·39-s + 6·41-s + 6·43-s − 2·45-s + 8·47-s + 9·49-s + 4·51-s + 6·53-s + 4·55-s + 4·57-s − 14·59-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s − 1.51·7-s + 1/3·9-s − 0.603·11-s − 0.554·13-s − 1.03·15-s + 0.485·17-s + 0.458·19-s − 1.74·21-s + 0.834·23-s − 1/5·25-s − 0.769·27-s − 0.696·33-s + 1.35·35-s + 1.64·37-s − 0.640·39-s + 0.937·41-s + 0.914·43-s − 0.298·45-s + 1.16·47-s + 9/7·49-s + 0.560·51-s + 0.824·53-s + 0.539·55-s + 0.529·57-s − 1.82·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 107648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.499104223\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.499104223\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 29 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 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
−13.63679835640419, −13.26045344552088, −12.70302172076592, −12.39951857056147, −11.83154992590539, −11.26586967597936, −10.65547205446508, −10.15498064050768, −9.563969015165020, −9.194786564951558, −8.931216134257417, −8.027779084773421, −7.717796397024200, −7.416442662625281, −6.831826233229159, −6.000206288039779, −5.731660455652094, −4.842083578583117, −4.196512089503447, −3.678822353255273, −3.205617829932160, −2.622528325387456, −2.438231231905445, −1.148441895016467, −0.3771416487391180,
0.3771416487391180, 1.148441895016467, 2.438231231905445, 2.622528325387456, 3.205617829932160, 3.678822353255273, 4.196512089503447, 4.842083578583117, 5.731660455652094, 6.000206288039779, 6.831826233229159, 7.416442662625281, 7.717796397024200, 8.027779084773421, 8.931216134257417, 9.194786564951558, 9.563969015165020, 10.15498064050768, 10.65547205446508, 11.26586967597936, 11.83154992590539, 12.39951857056147, 12.70302172076592, 13.26045344552088, 13.63679835640419
