| L(s) = 1 | + 3-s − 7-s + 9-s − 4·11-s − 6·13-s + 6·17-s − 4·19-s − 21-s − 8·23-s + 27-s + 10·29-s − 4·31-s − 4·33-s + 6·37-s − 6·39-s + 6·41-s + 4·43-s − 12·47-s + 49-s + 6·51-s − 6·53-s − 4·57-s − 4·59-s − 2·61-s − 63-s + 4·67-s − 8·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s − 1.20·11-s − 1.66·13-s + 1.45·17-s − 0.917·19-s − 0.218·21-s − 1.66·23-s + 0.192·27-s + 1.85·29-s − 0.718·31-s − 0.696·33-s + 0.986·37-s − 0.960·39-s + 0.937·41-s + 0.609·43-s − 1.75·47-s + 1/7·49-s + 0.840·51-s − 0.824·53-s − 0.529·57-s − 0.520·59-s − 0.256·61-s − 0.125·63-s + 0.488·67-s − 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.476174634\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.476174634\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| good | 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 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.ae |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 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
−15.95960029350645, −15.32525835317076, −14.62776082660587, −14.36587070654068, −13.82893287473191, −13.01791173947903, −12.49033852161387, −12.33122629054750, −11.50539686444932, −10.61287315889599, −10.10945226797866, −9.823425685748669, −9.213599813025955, −8.241601846749631, −7.856772311991409, −7.543089873796562, −6.602653101667155, −6.014434368183278, −5.211774520899051, −4.648544552595775, −3.921590986906025, −2.955445298253654, −2.599398206058364, −1.797972854274230, −0.4750528877381764,
0.4750528877381764, 1.797972854274230, 2.599398206058364, 2.955445298253654, 3.921590986906025, 4.648544552595775, 5.211774520899051, 6.014434368183278, 6.602653101667155, 7.543089873796562, 7.856772311991409, 8.241601846749631, 9.213599813025955, 9.823425685748669, 10.10945226797866, 10.61287315889599, 11.50539686444932, 12.33122629054750, 12.49033852161387, 13.01791173947903, 13.82893287473191, 14.36587070654068, 14.62776082660587, 15.32525835317076, 15.95960029350645
