| L(s) = 1 | − 3·5-s − 2·7-s + 6·11-s + 3·17-s − 2·19-s − 6·23-s + 4·25-s − 3·29-s + 4·31-s + 6·35-s − 7·37-s + 3·41-s + 10·43-s + 6·47-s − 3·49-s − 3·53-s − 18·55-s − 7·61-s + 10·67-s + 6·71-s − 13·73-s − 12·77-s + 4·79-s − 6·83-s − 9·85-s − 18·89-s + 6·95-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 0.755·7-s + 1.80·11-s + 0.727·17-s − 0.458·19-s − 1.25·23-s + 4/5·25-s − 0.557·29-s + 0.718·31-s + 1.01·35-s − 1.15·37-s + 0.468·41-s + 1.52·43-s + 0.875·47-s − 3/7·49-s − 0.412·53-s − 2.42·55-s − 0.896·61-s + 1.22·67-s + 0.712·71-s − 1.52·73-s − 1.36·77-s + 0.450·79-s − 0.658·83-s − 0.976·85-s − 1.90·89-s + 0.615·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 13 T + p T^{2} \) | 1.73.n |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−15.66403668578515, −15.27857914853164, −14.51063692148683, −14.19750435838001, −13.66830403637172, −12.69787728741412, −12.32149393015802, −12.01286920565306, −11.41986904745868, −10.92248878615196, −10.15891474035611, −9.578585865323397, −9.087740123570680, −8.438310916700379, −7.899770328466956, −7.263054493315534, −6.742747356652433, −6.136814681952603, −5.575664524194607, −4.407110634971458, −4.076574269388569, −3.593739141275329, −2.924229387735290, −1.832590659004001, −0.9224130541932181, 0,
0.9224130541932181, 1.832590659004001, 2.924229387735290, 3.593739141275329, 4.076574269388569, 4.407110634971458, 5.575664524194607, 6.136814681952603, 6.742747356652433, 7.263054493315534, 7.899770328466956, 8.438310916700379, 9.087740123570680, 9.578585865323397, 10.15891474035611, 10.92248878615196, 11.41986904745868, 12.01286920565306, 12.32149393015802, 12.69787728741412, 13.66830403637172, 14.19750435838001, 14.51063692148683, 15.27857914853164, 15.66403668578515
