| L(s) = 1 | + 5-s − 4·7-s + 11-s + 6·13-s + 2·17-s + 4·19-s + 8·23-s + 25-s + 6·29-s − 4·35-s + 2·37-s − 2·41-s + 8·43-s + 9·49-s − 2·53-s + 55-s + 4·59-s + 10·61-s + 6·65-s − 12·67-s + 16·71-s + 6·73-s − 4·77-s + 8·79-s − 16·83-s + 2·85-s − 10·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.51·7-s + 0.301·11-s + 1.66·13-s + 0.485·17-s + 0.917·19-s + 1.66·23-s + 1/5·25-s + 1.11·29-s − 0.676·35-s + 0.328·37-s − 0.312·41-s + 1.21·43-s + 9/7·49-s − 0.274·53-s + 0.134·55-s + 0.520·59-s + 1.28·61-s + 0.744·65-s − 1.46·67-s + 1.89·71-s + 0.702·73-s − 0.455·77-s + 0.900·79-s − 1.75·83-s + 0.216·85-s − 1.05·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.059573806\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.059573806\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−15.30144491341366, −14.33444960028435, −13.99962330296184, −13.41207298690456, −13.00968367773402, −12.58545191971236, −11.96399366138692, −11.20006470249082, −10.88874327897773, −10.04916143409167, −9.795536788311718, −9.072857317951304, −8.791295538500504, −8.060678024868615, −7.190683856424267, −6.756699376688373, −6.207535569103129, −5.753861387498853, −5.097987651159412, −4.221050171657134, −3.395111791508250, −3.206570881444579, −2.381808575325427, −1.171591738456993, −0.8002616240078163,
0.8002616240078163, 1.171591738456993, 2.381808575325427, 3.206570881444579, 3.395111791508250, 4.221050171657134, 5.097987651159412, 5.753861387498853, 6.207535569103129, 6.756699376688373, 7.190683856424267, 8.060678024868615, 8.791295538500504, 9.072857317951304, 9.795536788311718, 10.04916143409167, 10.88874327897773, 11.20006470249082, 11.96399366138692, 12.58545191971236, 13.00968367773402, 13.41207298690456, 13.99962330296184, 14.33444960028435, 15.30144491341366
