| L(s) = 1 | + 2·3-s + 2·5-s + 9-s + 2·11-s + 2·13-s + 4·15-s + 2·17-s + 2·19-s + 4·23-s − 25-s − 4·27-s + 6·29-s + 4·33-s − 10·37-s + 4·39-s + 6·41-s − 6·43-s + 2·45-s + 8·47-s + 4·51-s + 6·53-s + 4·55-s + 4·57-s + 14·59-s + 2·61-s + 4·65-s − 10·67-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-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 + 0.834·23-s − 1/5·25-s − 0.769·27-s + 1.11·29-s + 0.696·33-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 + 0.560·51-s + 0.824·53-s + 0.539·55-s + 0.529·57-s + 1.82·59-s + 0.256·61-s + 0.496·65-s − 1.22·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6272 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.153608433\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.153608433\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 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 |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 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.ao |
| 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.i |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 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
−8.241245270225530562998002812161, −7.34525112137563992258201860959, −6.75316825864606352254488002562, −5.84930432399668081424677432212, −5.32686527890757944440591569596, −4.22305701392233207301956082818, −3.45578431405321998061897801865, −2.78812977513684531419313254549, −1.94432221551497566565336210098, −1.07814716423299698059230098817,
1.07814716423299698059230098817, 1.94432221551497566565336210098, 2.78812977513684531419313254549, 3.45578431405321998061897801865, 4.22305701392233207301956082818, 5.32686527890757944440591569596, 5.84930432399668081424677432212, 6.75316825864606352254488002562, 7.34525112137563992258201860959, 8.241245270225530562998002812161
