| L(s) = 1 | − 5-s − 4·7-s − 2·13-s − 2·17-s − 4·19-s + 25-s − 6·29-s + 8·31-s + 4·35-s + 6·37-s − 10·41-s + 43-s + 9·49-s − 6·53-s − 2·61-s + 2·65-s − 12·67-s + 8·71-s − 2·73-s − 8·79-s − 4·83-s + 2·85-s + 6·89-s + 8·91-s + 4·95-s + 10·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.51·7-s − 0.554·13-s − 0.485·17-s − 0.917·19-s + 1/5·25-s − 1.11·29-s + 1.43·31-s + 0.676·35-s + 0.986·37-s − 1.56·41-s + 0.152·43-s + 9/7·49-s − 0.824·53-s − 0.256·61-s + 0.248·65-s − 1.46·67-s + 0.949·71-s − 0.234·73-s − 0.900·79-s − 0.439·83-s + 0.216·85-s + 0.635·89-s + 0.838·91-s + 0.410·95-s + 1.01·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 43 | \( 1 - T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−13.83379344412224, −13.42822830108081, −13.06304297355426, −12.59934784177682, −12.14242629533133, −11.69931479772779, −11.11528632501451, −10.57173028451603, −10.09847828944040, −9.658532877696353, −9.191092847588718, −8.704481271597620, −8.085314693585033, −7.626514850439969, −6.942136349528516, −6.581967762302291, −6.178921470933549, −5.559567190415269, −4.822885001743122, −4.302814652822397, −3.800074302271253, −3.127969955110284, −2.711344576896225, −2.043877683945854, −1.133183121356829, 0, 0,
1.133183121356829, 2.043877683945854, 2.711344576896225, 3.127969955110284, 3.800074302271253, 4.302814652822397, 4.822885001743122, 5.559567190415269, 6.178921470933549, 6.581967762302291, 6.942136349528516, 7.626514850439969, 8.085314693585033, 8.704481271597620, 9.191092847588718, 9.658532877696353, 10.09847828944040, 10.57173028451603, 11.11528632501451, 11.69931479772779, 12.14242629533133, 12.59934784177682, 13.06304297355426, 13.42822830108081, 13.83379344412224
