| L(s) = 1 | − 3·5-s − 4·7-s + 3·13-s − 3·17-s − 4·19-s − 8·23-s + 4·25-s + 5·29-s + 4·31-s + 12·35-s + 11·37-s − 7·41-s + 12·43-s − 8·47-s + 9·49-s + 53-s − 4·59-s − 2·61-s − 9·65-s − 4·67-s + 12·71-s + 10·73-s − 8·79-s + 4·83-s + 9·85-s − 3·89-s − 12·91-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 1.51·7-s + 0.832·13-s − 0.727·17-s − 0.917·19-s − 1.66·23-s + 4/5·25-s + 0.928·29-s + 0.718·31-s + 2.02·35-s + 1.80·37-s − 1.09·41-s + 1.82·43-s − 1.16·47-s + 9/7·49-s + 0.137·53-s − 0.520·59-s − 0.256·61-s − 1.11·65-s − 0.488·67-s + 1.42·71-s + 1.17·73-s − 0.900·79-s + 0.439·83-s + 0.976·85-s − 0.317·89-s − 1.25·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17424 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17424 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 11 T + p T^{2} \) | 1.37.al |
| 41 | \( 1 + 7 T + p T^{2} \) | 1.41.h |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 97 | \( 1 + 13 T + p T^{2} \) | 1.97.n |
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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
−15.95895401032294, −15.61667268930346, −15.37597956472363, −14.51095021629568, −13.83767387064870, −13.32616094358348, −12.66527081640536, −12.37923148739063, −11.65497236577509, −11.20604502693789, −10.53372615731414, −9.958032877184772, −9.377665261565610, −8.614518369763991, −8.181149841934773, −7.625858328008450, −6.771985646959634, −6.295642947675732, −5.949748943528739, −4.647249707877406, −4.163563364434158, −3.645309210712431, −2.963116743295113, −2.161108003256668, −0.7756358043250667, 0,
0.7756358043250667, 2.161108003256668, 2.963116743295113, 3.645309210712431, 4.163563364434158, 4.647249707877406, 5.949748943528739, 6.295642947675732, 6.771985646959634, 7.625858328008450, 8.181149841934773, 8.614518369763991, 9.377665261565610, 9.958032877184772, 10.53372615731414, 11.20604502693789, 11.65497236577509, 12.37923148739063, 12.66527081640536, 13.32616094358348, 13.83767387064870, 14.51095021629568, 15.37597956472363, 15.61667268930346, 15.95895401032294
