| L(s) = 1 | − 3-s − 7-s + 9-s − 11-s − 4·13-s − 4·17-s − 2·19-s + 21-s − 2·23-s − 27-s + 6·29-s − 6·31-s + 33-s + 6·37-s + 4·39-s − 8·41-s − 43-s + 9·47-s + 49-s + 4·51-s + 13·53-s + 2·57-s + 10·59-s + 10·61-s − 63-s + 4·67-s + 2·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.10·13-s − 0.970·17-s − 0.458·19-s + 0.218·21-s − 0.417·23-s − 0.192·27-s + 1.11·29-s − 1.07·31-s + 0.174·33-s + 0.986·37-s + 0.640·39-s − 1.24·41-s − 0.152·43-s + 1.31·47-s + 1/7·49-s + 0.560·51-s + 1.78·53-s + 0.264·57-s + 1.30·59-s + 1.28·61-s − 0.125·63-s + 0.488·67-s + 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.383666548\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.383666548\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| good | 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 - 13 T + p T^{2} \) | 1.53.an |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 5 T + p T^{2} \) | 1.73.f |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - T + p T^{2} \) | 1.83.ab |
| 89 | \( 1 - T + p T^{2} \) | 1.89.ab |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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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
−12.57377066268231, −11.87484316034732, −11.73513696983882, −11.23475659792320, −10.52273467528453, −10.32030747855046, −9.955344752022614, −9.331128358951171, −8.925924087692098, −8.393923276615511, −7.940994050366679, −7.241986259220656, −6.992901601785737, −6.525933447323653, −6.040919028952056, −5.322137576143591, −5.204875769430029, −4.455914239857413, −4.070843550928177, −3.552674713638999, −2.696622084311266, −2.322807854776017, −1.875360212836094, −0.8217193923765733, −0.4023954479969369,
0.4023954479969369, 0.8217193923765733, 1.875360212836094, 2.322807854776017, 2.696622084311266, 3.552674713638999, 4.070843550928177, 4.455914239857413, 5.204875769430029, 5.322137576143591, 6.040919028952056, 6.525933447323653, 6.992901601785737, 7.241986259220656, 7.940994050366679, 8.393923276615511, 8.925924087692098, 9.331128358951171, 9.955344752022614, 10.32030747855046, 10.52273467528453, 11.23475659792320, 11.73513696983882, 11.87484316034732, 12.57377066268231
