| L(s) = 1 | + 5-s − 2·13-s − 6·17-s + 4·19-s − 8·23-s + 25-s − 2·29-s + 4·31-s − 10·37-s − 2·41-s + 4·43-s − 8·47-s − 7·49-s − 2·53-s + 8·59-s + 2·61-s − 2·65-s + 12·67-s − 8·71-s − 14·73-s − 12·79-s − 4·83-s − 6·85-s + 14·89-s + 4·95-s + 2·97-s − 10·101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.554·13-s − 1.45·17-s + 0.917·19-s − 1.66·23-s + 1/5·25-s − 0.371·29-s + 0.718·31-s − 1.64·37-s − 0.312·41-s + 0.609·43-s − 1.16·47-s − 49-s − 0.274·53-s + 1.04·59-s + 0.256·61-s − 0.248·65-s + 1.46·67-s − 0.949·71-s − 1.63·73-s − 1.35·79-s − 0.439·83-s − 0.650·85-s + 1.48·89-s + 0.410·95-s + 0.203·97-s − 0.995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−8.466905627005947042329099996565, −7.62711950498123032171918638060, −6.83243352642680668164749557491, −6.15740871209984297437683595812, −5.27792996915882287403209471728, −4.54126059396742331170753302700, −3.57512110223300924839054817872, −2.49396557183660641497783794085, −1.66014780933081759283646218290, 0,
1.66014780933081759283646218290, 2.49396557183660641497783794085, 3.57512110223300924839054817872, 4.54126059396742331170753302700, 5.27792996915882287403209471728, 6.15740871209984297437683595812, 6.83243352642680668164749557491, 7.62711950498123032171918638060, 8.466905627005947042329099996565
