| L(s) = 1 | − 5-s + 2·7-s + 2·11-s + 2·13-s − 4·17-s − 4·19-s + 4·23-s + 25-s + 6·29-s + 4·31-s − 2·35-s − 4·37-s − 43-s + 12·47-s − 3·49-s + 6·53-s − 2·55-s − 6·59-s + 8·61-s − 2·65-s − 4·67-s − 16·71-s − 14·73-s + 4·77-s − 10·83-s + 4·85-s − 2·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.755·7-s + 0.603·11-s + 0.554·13-s − 0.970·17-s − 0.917·19-s + 0.834·23-s + 1/5·25-s + 1.11·29-s + 0.718·31-s − 0.338·35-s − 0.657·37-s − 0.152·43-s + 1.75·47-s − 3/7·49-s + 0.824·53-s − 0.269·55-s − 0.781·59-s + 1.02·61-s − 0.248·65-s − 0.488·67-s − 1.89·71-s − 1.63·73-s + 0.455·77-s − 1.09·83-s + 0.433·85-s − 0.211·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 43 | \( 1 + T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 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.ai |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 10 T + p T^{2} \) | 1.83.k |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−14.60420123921642, −14.01097085467484, −13.55022147090001, −13.07796143982099, −12.44048874577878, −11.91557905514114, −11.52927368000300, −11.00500745533957, −10.52744942016452, −10.13653472540710, −9.139471617947070, −8.831987522835220, −8.425004373301464, −7.920413989953285, −7.045394961708929, −6.869119733591879, −6.135516291190078, −5.578801342102145, −4.742440965207171, −4.363577982351026, −3.941268271907867, −3.023283266105648, −2.482800064085579, −1.587053255528691, −1.039502169377562, 0,
1.039502169377562, 1.587053255528691, 2.482800064085579, 3.023283266105648, 3.941268271907867, 4.363577982351026, 4.742440965207171, 5.578801342102145, 6.135516291190078, 6.869119733591879, 7.045394961708929, 7.920413989953285, 8.425004373301464, 8.831987522835220, 9.139471617947070, 10.13653472540710, 10.52744942016452, 11.00500745533957, 11.52927368000300, 11.91557905514114, 12.44048874577878, 13.07796143982099, 13.55022147090001, 14.01097085467484, 14.60420123921642
