| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s + 4·11-s + 6·13-s − 15-s + 6·17-s + 8·19-s − 21-s − 8·23-s + 25-s + 27-s + 2·29-s + 4·33-s + 35-s − 2·37-s + 6·39-s − 2·41-s − 8·43-s − 45-s − 12·47-s + 49-s + 6·51-s − 53-s − 4·55-s + 8·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s + 1.66·13-s − 0.258·15-s + 1.45·17-s + 1.83·19-s − 0.218·21-s − 1.66·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.696·33-s + 0.169·35-s − 0.328·37-s + 0.960·39-s − 0.312·41-s − 1.21·43-s − 0.149·45-s − 1.75·47-s + 1/7·49-s + 0.840·51-s − 0.137·53-s − 0.539·55-s + 1.05·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 356160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 356160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 53 | \( 1 + T \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−12.65361707031184, −12.19771448628813, −11.90144319266260, −11.53247304091018, −11.04082090558681, −10.37476118507724, −9.932302431194460, −9.557762910930121, −9.237506149621002, −8.452826912410214, −8.288009539680527, −7.809349823182407, −7.329047830351295, −6.649299529646740, −6.400105051190315, −5.785867304263293, −5.350874269229228, −4.675421332358216, −3.919946008645871, −3.712296623525093, −3.239512653428031, −2.936372087232846, −1.810066701839829, −1.366266690382372, −1.021743635816343, 0,
1.021743635816343, 1.366266690382372, 1.810066701839829, 2.936372087232846, 3.239512653428031, 3.712296623525093, 3.919946008645871, 4.675421332358216, 5.350874269229228, 5.785867304263293, 6.400105051190315, 6.649299529646740, 7.329047830351295, 7.809349823182407, 8.288009539680527, 8.452826912410214, 9.237506149621002, 9.557762910930121, 9.932302431194460, 10.37476118507724, 11.04082090558681, 11.53247304091018, 11.90144319266260, 12.19771448628813, 12.65361707031184
