| L(s) = 1 | − 2-s + 4-s − 5-s + 7-s − 8-s + 10-s − 13-s − 14-s + 16-s − 5·19-s − 20-s − 6·23-s + 25-s + 26-s + 28-s − 7·31-s − 32-s − 35-s + 8·37-s + 5·38-s + 40-s − 6·41-s + 7·43-s + 6·46-s − 3·47-s − 6·49-s − 50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s − 0.353·8-s + 0.316·10-s − 0.277·13-s − 0.267·14-s + 1/4·16-s − 1.14·19-s − 0.223·20-s − 1.25·23-s + 1/5·25-s + 0.196·26-s + 0.188·28-s − 1.25·31-s − 0.176·32-s − 0.169·35-s + 1.31·37-s + 0.811·38-s + 0.158·40-s − 0.937·41-s + 1.06·43-s + 0.884·46-s − 0.437·47-s − 6/7·49-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4004634173\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4004634173\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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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
−13.34326581943221, −12.71600903818663, −12.46389921837856, −11.87872700766822, −11.36036290776320, −11.01393311145468, −10.55467280565347, −9.975910748264703, −9.578893607603830, −8.974654317232225, −8.492394002229451, −8.111314751362079, −7.497414760786826, −7.308141825107679, −6.450957623384222, −6.095659422516717, −5.567410233109568, −4.670517200960896, −4.430115484626160, −3.670909924054027, −3.140787640772169, −2.295428389201823, −1.916441191231684, −1.185945753917733, −0.2172282181822617,
0.2172282181822617, 1.185945753917733, 1.916441191231684, 2.295428389201823, 3.140787640772169, 3.670909924054027, 4.430115484626160, 4.670517200960896, 5.567410233109568, 6.095659422516717, 6.450957623384222, 7.308141825107679, 7.497414760786826, 8.111314751362079, 8.492394002229451, 8.974654317232225, 9.578893607603830, 9.975910748264703, 10.55467280565347, 11.01393311145468, 11.36036290776320, 11.87872700766822, 12.46389921837856, 12.71600903818663, 13.34326581943221
