| L(s) = 1 | + 3-s + 7-s + 9-s + 4·11-s − 2·13-s + 6·17-s + 4·19-s + 21-s + 8·23-s + 27-s + 2·29-s + 4·33-s − 2·37-s − 2·39-s + 10·41-s − 4·43-s + 49-s + 6·51-s + 14·53-s + 4·57-s + 12·59-s + 2·61-s + 63-s + 4·67-s + 8·69-s − 2·73-s + 4·77-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 1.45·17-s + 0.917·19-s + 0.218·21-s + 1.66·23-s + 0.192·27-s + 0.371·29-s + 0.696·33-s − 0.328·37-s − 0.320·39-s + 1.56·41-s − 0.609·43-s + 1/7·49-s + 0.840·51-s + 1.92·53-s + 0.529·57-s + 1.56·59-s + 0.256·61-s + 0.125·63-s + 0.488·67-s + 0.963·69-s − 0.234·73-s + 0.455·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.638723591\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.638723591\) |
| \(L(\frac{3}{2})\) |
|
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.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 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.ai |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−14.92666157959846, −14.44412244842656, −14.13656611232191, −13.49301751092356, −12.95425296121785, −12.20957109544366, −11.97231072380504, −11.35884327638103, −10.74399426514283, −10.03741311548107, −9.595208846433967, −9.118974160571965, −8.552675387618798, −7.964652130735078, −7.258994130591628, −7.038753806250300, −6.236713336525412, −5.321783696644939, −5.122969210343696, −4.119514961461019, −3.673414616428152, −2.945543206636164, −2.340145401383753, −1.233537064633089, −0.9537892464135103,
0.9537892464135103, 1.233537064633089, 2.340145401383753, 2.945543206636164, 3.673414616428152, 4.119514961461019, 5.122969210343696, 5.321783696644939, 6.236713336525412, 7.038753806250300, 7.258994130591628, 7.964652130735078, 8.552675387618798, 9.118974160571965, 9.595208846433967, 10.03741311548107, 10.74399426514283, 11.35884327638103, 11.97231072380504, 12.20957109544366, 12.95425296121785, 13.49301751092356, 14.13656611232191, 14.44412244842656, 14.92666157959846
