| L(s) = 1 | − 3-s + 5-s − 7-s + 9-s + 4·11-s − 15-s + 6·17-s + 8·19-s + 21-s + 4·23-s + 25-s − 27-s − 2·29-s − 4·31-s − 4·33-s − 35-s − 10·37-s + 2·41-s − 8·43-s + 45-s + 8·47-s + 49-s − 6·51-s + 6·53-s + 4·55-s − 8·57-s + 8·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.258·15-s + 1.45·17-s + 1.83·19-s + 0.218·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.718·31-s − 0.696·33-s − 0.169·35-s − 1.64·37-s + 0.312·41-s − 1.21·43-s + 0.149·45-s + 1.16·47-s + 1/7·49-s − 0.840·51-s + 0.824·53-s + 0.539·55-s − 1.05·57-s + 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 283920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 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.g |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.04519597206535, −12.26342833530177, −12.06778996314699, −11.71290375588358, −11.23840027739654, −10.58575996994993, −10.17704159309400, −9.792243168466355, −9.355559668088150, −8.869538179536310, −8.531763566656359, −7.535096402759712, −7.259747651879622, −7.029912879084032, −6.276941928427544, −5.691774422556649, −5.591834392404889, −4.970942003686707, −4.390623885798073, −3.586542751320182, −3.379828988145663, −2.818077435570815, −1.847790782711235, −1.284195685329704, −0.9849371566951925, 0,
0.9849371566951925, 1.284195685329704, 1.847790782711235, 2.818077435570815, 3.379828988145663, 3.586542751320182, 4.390623885798073, 4.970942003686707, 5.591834392404889, 5.691774422556649, 6.276941928427544, 7.029912879084032, 7.259747651879622, 7.535096402759712, 8.531763566656359, 8.869538179536310, 9.355559668088150, 9.792243168466355, 10.17704159309400, 10.58575996994993, 11.23840027739654, 11.71290375588358, 12.06778996314699, 12.26342833530177, 13.04519597206535
