| L(s) = 1 | − 5-s − 2·7-s − 2·13-s − 6·17-s + 25-s − 4·29-s + 8·31-s + 2·35-s − 6·37-s − 6·43-s − 8·47-s − 3·49-s + 14·53-s − 4·59-s − 8·61-s + 2·65-s − 12·67-s − 12·71-s + 2·73-s + 16·79-s − 6·83-s + 6·85-s + 2·89-s + 4·91-s + 6·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s − 0.554·13-s − 1.45·17-s + 1/5·25-s − 0.742·29-s + 1.43·31-s + 0.338·35-s − 0.986·37-s − 0.914·43-s − 1.16·47-s − 3/7·49-s + 1.92·53-s − 0.520·59-s − 1.02·61-s + 0.248·65-s − 1.46·67-s − 1.42·71-s + 0.234·73-s + 1.80·79-s − 0.658·83-s + 0.650·85-s + 0.211·89-s + 0.419·91-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 174240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 174240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 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
−13.51778737334880, −13.21843709303116, −12.82237938833081, −12.12538207168657, −11.82294593376901, −11.45699787764168, −10.72532672979120, −10.37484016007377, −9.951205566584720, −9.198036113570779, −9.078101051655815, −8.350510744789961, −7.994790935398854, −7.196340535451322, −7.015084541150116, −6.324301552887496, −6.074178742437456, −5.180742549576832, −4.748557070402104, −4.286909349730204, −3.605226873712952, −3.136402568617411, −2.525258698718031, −1.937430162752152, −1.148549792969092, 0, 0,
1.148549792969092, 1.937430162752152, 2.525258698718031, 3.136402568617411, 3.605226873712952, 4.286909349730204, 4.748557070402104, 5.180742549576832, 6.074178742437456, 6.324301552887496, 7.015084541150116, 7.196340535451322, 7.994790935398854, 8.350510744789961, 9.078101051655815, 9.198036113570779, 9.951205566584720, 10.37484016007377, 10.72532672979120, 11.45699787764168, 11.82294593376901, 12.12538207168657, 12.82237938833081, 13.21843709303116, 13.51778737334880
