| L(s) = 1 | − 2-s + 4-s + 4·7-s − 8-s + 2·11-s − 6·13-s − 4·14-s + 16-s − 8·19-s − 2·22-s + 6·26-s + 4·28-s − 2·29-s − 8·31-s − 32-s − 10·37-s + 8·38-s + 6·41-s − 4·43-s + 2·44-s + 6·47-s + 9·49-s − 6·52-s + 2·53-s − 4·56-s + 2·58-s − 12·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.51·7-s − 0.353·8-s + 0.603·11-s − 1.66·13-s − 1.06·14-s + 1/4·16-s − 1.83·19-s − 0.426·22-s + 1.17·26-s + 0.755·28-s − 0.371·29-s − 1.43·31-s − 0.176·32-s − 1.64·37-s + 1.29·38-s + 0.937·41-s − 0.609·43-s + 0.301·44-s + 0.875·47-s + 9/7·49-s − 0.832·52-s + 0.274·53-s − 0.534·56-s + 0.262·58-s − 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.136397976\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.136397976\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 79 | \( 1 - T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−14.85154157892717, −14.54226819760804, −14.19671425595091, −13.35204515257706, −12.57451577073024, −12.25198503665451, −11.72440606951207, −11.11686968044080, −10.65645488500863, −10.29466800308032, −9.450492017742760, −8.976517310171205, −8.581486386269321, −7.804202832176684, −7.537587723982720, −6.884097401715728, −6.309624705615293, −5.405222156511117, −5.004210906254391, −4.299360605541824, −3.718534919110848, −2.601464887590437, −1.979340119495618, −1.623501660585066, −0.4275439434428177,
0.4275439434428177, 1.623501660585066, 1.979340119495618, 2.601464887590437, 3.718534919110848, 4.299360605541824, 5.004210906254391, 5.405222156511117, 6.309624705615293, 6.884097401715728, 7.537587723982720, 7.804202832176684, 8.581486386269321, 8.976517310171205, 9.450492017742760, 10.29466800308032, 10.65645488500863, 11.11686968044080, 11.72440606951207, 12.25198503665451, 12.57451577073024, 13.35204515257706, 14.19671425595091, 14.54226819760804, 14.85154157892717
