| L(s) = 1 | + 2·3-s + 7-s + 9-s − 4·13-s − 6·17-s + 2·19-s + 2·21-s − 4·27-s + 6·29-s + 4·31-s + 2·37-s − 8·39-s + 6·41-s − 8·43-s − 12·47-s + 49-s − 12·51-s + 6·53-s + 4·57-s − 6·59-s − 8·61-s + 63-s + 4·67-s − 2·73-s − 8·79-s − 11·81-s + 6·83-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.377·7-s + 1/3·9-s − 1.10·13-s − 1.45·17-s + 0.458·19-s + 0.436·21-s − 0.769·27-s + 1.11·29-s + 0.718·31-s + 0.328·37-s − 1.28·39-s + 0.937·41-s − 1.21·43-s − 1.75·47-s + 1/7·49-s − 1.68·51-s + 0.824·53-s + 0.529·57-s − 0.781·59-s − 1.02·61-s + 0.125·63-s + 0.488·67-s − 0.234·73-s − 0.900·79-s − 1.22·81-s + 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 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 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 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.c |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−16.85028860612941, −15.97963592665803, −15.52663895442357, −14.91851121803239, −14.52470611495479, −13.98610838864149, −13.39957657643284, −13.00388505079707, −12.08274606587342, −11.68286818225714, −10.96079833821916, −10.25101830002950, −9.571598304181429, −9.175222764276347, −8.357959430378233, −8.112181817195535, −7.315233977606714, −6.731904705199390, −5.958936603332825, −4.893659077922679, −4.573857446042448, −3.637501930074888, −2.762266670183917, −2.390131236528301, −1.431304142554610, 0,
1.431304142554610, 2.390131236528301, 2.762266670183917, 3.637501930074888, 4.573857446042448, 4.893659077922679, 5.958936603332825, 6.731904705199390, 7.315233977606714, 8.112181817195535, 8.357959430378233, 9.175222764276347, 9.571598304181429, 10.25101830002950, 10.96079833821916, 11.68286818225714, 12.08274606587342, 13.00388505079707, 13.39957657643284, 13.98610838864149, 14.52470611495479, 14.91851121803239, 15.52663895442357, 15.97963592665803, 16.85028860612941
