| L(s) = 1 | − 5-s + 2·7-s + 2·11-s + 5·13-s + 2·17-s + 6·19-s − 4·25-s − 5·29-s − 2·35-s − 2·37-s + 9·41-s − 8·43-s − 10·47-s − 3·49-s − 9·53-s − 2·55-s − 10·59-s + 13·61-s − 5·65-s − 8·67-s + 2·71-s + 3·73-s + 4·77-s + 12·83-s − 2·85-s − 7·89-s + 10·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.755·7-s + 0.603·11-s + 1.38·13-s + 0.485·17-s + 1.37·19-s − 4/5·25-s − 0.928·29-s − 0.338·35-s − 0.328·37-s + 1.40·41-s − 1.21·43-s − 1.45·47-s − 3/7·49-s − 1.23·53-s − 0.269·55-s − 1.30·59-s + 1.66·61-s − 0.620·65-s − 0.977·67-s + 0.237·71-s + 0.351·73-s + 0.455·77-s + 1.31·83-s − 0.216·85-s − 0.741·89-s + 1.04·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 23 | \( 1 \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 13 T + p T^{2} \) | 1.61.an |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 7 T + p T^{2} \) | 1.89.h |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
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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
−12.79346233193166, −12.47970165185813, −11.75678498546103, −11.52568212760554, −11.20645707254582, −10.82770627392451, −10.10237195268871, −9.626319169323060, −9.279550795143040, −8.714520565693816, −8.182042061967078, −7.781657401021367, −7.571879145528361, −6.754564827517153, −6.381192221291052, −5.816764527011868, −5.294118877168904, −4.905005302021882, −4.155166400488248, −3.769206685452597, −3.338263049097318, −2.794728393934411, −1.709995067426943, −1.553816485923515, −0.9004722393414709, 0,
0.9004722393414709, 1.553816485923515, 1.709995067426943, 2.794728393934411, 3.338263049097318, 3.769206685452597, 4.155166400488248, 4.905005302021882, 5.294118877168904, 5.816764527011868, 6.381192221291052, 6.754564827517153, 7.571879145528361, 7.781657401021367, 8.182042061967078, 8.714520565693816, 9.279550795143040, 9.626319169323060, 10.10237195268871, 10.82770627392451, 11.20645707254582, 11.52568212760554, 11.75678498546103, 12.47970165185813, 12.79346233193166
