| L(s) = 1 | − 2·3-s − 2·5-s + 9-s + 4·15-s − 6·17-s + 6·19-s + 6·23-s − 25-s + 4·27-s − 6·29-s − 4·31-s − 2·37-s + 2·41-s − 6·43-s − 2·45-s + 6·47-s − 7·49-s + 12·51-s + 10·53-s − 12·57-s + 6·59-s + 6·61-s + 2·67-s − 12·69-s − 6·71-s − 2·73-s + 2·75-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s + 1/3·9-s + 1.03·15-s − 1.45·17-s + 1.37·19-s + 1.25·23-s − 1/5·25-s + 0.769·27-s − 1.11·29-s − 0.718·31-s − 0.328·37-s + 0.312·41-s − 0.914·43-s − 0.298·45-s + 0.875·47-s − 49-s + 1.68·51-s + 1.37·53-s − 1.58·57-s + 0.781·59-s + 0.768·61-s + 0.244·67-s − 1.44·69-s − 0.712·71-s − 0.234·73-s + 0.230·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 305552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 305552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| 113 | \( 1 - T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−12.96454228986451, −12.24970178887115, −11.77756478936868, −11.55961689316610, −11.22816044845760, −10.75847677821653, −10.38578923704863, −9.708071203249925, −9.143306119573817, −8.866453125114458, −8.243177254620091, −7.685708870093600, −7.199172356383684, −6.842572770907685, −6.447793499903434, −5.632950570220363, −5.348066413693383, −5.022488591583543, −4.186795750233501, −4.008754683107736, −3.237814958801443, −2.753658440399780, −1.963588476708996, −1.245652348244837, −0.5559637928197737, 0,
0.5559637928197737, 1.245652348244837, 1.963588476708996, 2.753658440399780, 3.237814958801443, 4.008754683107736, 4.186795750233501, 5.022488591583543, 5.348066413693383, 5.632950570220363, 6.447793499903434, 6.842572770907685, 7.199172356383684, 7.685708870093600, 8.243177254620091, 8.866453125114458, 9.143306119573817, 9.708071203249925, 10.38578923704863, 10.75847677821653, 11.22816044845760, 11.55961689316610, 11.77756478936868, 12.24970178887115, 12.96454228986451
