| L(s) = 1 | − 4·7-s + 4·11-s + 13-s − 4·17-s + 6·19-s − 5·25-s + 6·29-s + 4·31-s + 6·37-s + 6·41-s − 4·43-s − 8·47-s + 9·49-s − 6·53-s + 12·59-s + 10·61-s − 2·67-s − 12·71-s − 10·73-s − 16·77-s − 10·79-s + 4·83-s − 10·89-s − 4·91-s + 14·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 1.20·11-s + 0.277·13-s − 0.970·17-s + 1.37·19-s − 25-s + 1.11·29-s + 0.718·31-s + 0.986·37-s + 0.937·41-s − 0.609·43-s − 1.16·47-s + 9/7·49-s − 0.824·53-s + 1.56·59-s + 1.28·61-s − 0.244·67-s − 1.42·71-s − 1.17·73-s − 1.82·77-s − 1.12·79-s + 0.439·83-s − 1.05·89-s − 0.419·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14976 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.722188737\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.722188737\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 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
−16.03208765022513, −15.80143887752674, −14.92251578792338, −14.37313127258940, −13.74363573370334, −13.22853615181774, −12.87906427258556, −11.94302451980890, −11.71866946241061, −11.10135507984234, −10.08212348458565, −9.821511026325105, −9.295012728910753, −8.696073084409415, −7.998062576743343, −7.116365132968853, −6.644618457157043, −6.178218270648005, −5.559313456727462, −4.496687545612345, −3.980295847531562, −3.204715626059907, −2.670863323228263, −1.510967449248098, −0.5926616672340666,
0.5926616672340666, 1.510967449248098, 2.670863323228263, 3.204715626059907, 3.980295847531562, 4.496687545612345, 5.559313456727462, 6.178218270648005, 6.644618457157043, 7.116365132968853, 7.998062576743343, 8.696073084409415, 9.295012728910753, 9.821511026325105, 10.08212348458565, 11.10135507984234, 11.71866946241061, 11.94302451980890, 12.87906427258556, 13.22853615181774, 13.74363573370334, 14.37313127258940, 14.92251578792338, 15.80143887752674, 16.03208765022513
