| L(s) = 1 | − 3-s + 2·7-s − 2·9-s + 5·11-s + 2·13-s + 3·17-s − 19-s − 2·21-s − 2·23-s + 5·27-s − 8·31-s − 5·33-s + 6·37-s − 2·39-s − 12·41-s + 43-s − 10·47-s − 3·49-s − 3·51-s − 10·53-s + 57-s + 4·59-s + 5·61-s − 4·63-s − 4·67-s + 2·69-s − 71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s − 2/3·9-s + 1.50·11-s + 0.554·13-s + 0.727·17-s − 0.229·19-s − 0.436·21-s − 0.417·23-s + 0.962·27-s − 1.43·31-s − 0.870·33-s + 0.986·37-s − 0.320·39-s − 1.87·41-s + 0.152·43-s − 1.45·47-s − 3/7·49-s − 0.420·51-s − 1.37·53-s + 0.132·57-s + 0.520·59-s + 0.640·61-s − 0.503·63-s − 0.488·67-s + 0.240·69-s − 0.118·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 229 | \( 1 - T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + T + p T^{2} \) | 1.71.b |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 11 T + p T^{2} \) | 1.83.l |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 - 9 T + p T^{2} \) | 1.97.aj |
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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
−14.28634696337169, −13.77151128598438, −12.92315233408478, −12.64758593634317, −11.94191829211205, −11.48384483032731, −11.32993677663499, −10.89407139115071, −9.998716758075215, −9.802599077317579, −8.941184884279370, −8.656055553589215, −8.122392474496801, −7.573897188833572, −6.855107805590827, −6.363447819655395, −5.975313261532267, −5.357273807322022, −4.844113797682347, −4.269623869583062, −3.474452166287194, −3.264444834796321, −2.077918677193293, −1.588991186228221, −0.9430078677251465, 0,
0.9430078677251465, 1.588991186228221, 2.077918677193293, 3.264444834796321, 3.474452166287194, 4.269623869583062, 4.844113797682347, 5.357273807322022, 5.975313261532267, 6.363447819655395, 6.855107805590827, 7.573897188833572, 8.122392474496801, 8.656055553589215, 8.941184884279370, 9.802599077317579, 9.998716758075215, 10.89407139115071, 11.32993677663499, 11.48384483032731, 11.94191829211205, 12.64758593634317, 12.92315233408478, 13.77151128598438, 14.28634696337169
