| L(s) = 1 | − 2-s + 4-s − 5·7-s − 8-s − 11-s − 2·13-s + 5·14-s + 16-s − 4·17-s − 2·19-s + 22-s + 4·23-s + 2·26-s − 5·28-s + 29-s − 8·31-s − 32-s + 4·34-s − 2·37-s + 2·38-s + 5·41-s + 13·43-s − 44-s − 4·46-s − 3·47-s + 18·49-s − 2·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.88·7-s − 0.353·8-s − 0.301·11-s − 0.554·13-s + 1.33·14-s + 1/4·16-s − 0.970·17-s − 0.458·19-s + 0.213·22-s + 0.834·23-s + 0.392·26-s − 0.944·28-s + 0.185·29-s − 1.43·31-s − 0.176·32-s + 0.685·34-s − 0.328·37-s + 0.324·38-s + 0.780·41-s + 1.98·43-s − 0.150·44-s − 0.589·46-s − 0.437·47-s + 18/7·49-s − 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6028864423\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6028864423\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| 29 | \( 1 - T \) | |
| good | 7 | \( 1 + 5 T + p T^{2} \) | 1.7.f |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 - 13 T + p T^{2} \) | 1.43.an |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 9 T + p T^{2} \) | 1.67.aj |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 - 13 T + p T^{2} \) | 1.79.an |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
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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
−13.17770064340280, −12.84155760536205, −12.39825451122608, −12.23176442245431, −11.13963140948934, −10.91206575812352, −10.60674273608564, −9.800618097467709, −9.447140739230395, −9.248440358382670, −8.693595908174417, −8.054512851515538, −7.326018457580959, −7.158440480798794, −6.363798509310579, −6.315780225939565, −5.524717795133615, −4.943470485510413, −4.178865130585772, −3.577146787344500, −3.094213902579542, −2.386837312035684, −2.111763937396036, −0.9169347445248217, −0.3114126322442234,
0.3114126322442234, 0.9169347445248217, 2.111763937396036, 2.386837312035684, 3.094213902579542, 3.577146787344500, 4.178865130585772, 4.943470485510413, 5.524717795133615, 6.315780225939565, 6.363798509310579, 7.158440480798794, 7.326018457580959, 8.054512851515538, 8.693595908174417, 9.248440358382670, 9.447140739230395, 9.800618097467709, 10.60674273608564, 10.91206575812352, 11.13963140948934, 12.23176442245431, 12.39825451122608, 12.84155760536205, 13.17770064340280
