| L(s) = 1 | − 3·5-s + 2·7-s + 11-s − 3·13-s + 4·17-s − 8·19-s + 4·25-s − 29-s − 3·31-s − 6·35-s + 8·37-s + 2·41-s + 7·43-s + 11·47-s − 3·49-s + 53-s − 3·55-s + 4·59-s − 4·61-s + 9·65-s − 4·67-s − 2·71-s − 12·73-s + 2·77-s + 7·79-s − 12·85-s + 6·89-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.755·7-s + 0.301·11-s − 0.832·13-s + 0.970·17-s − 1.83·19-s + 4/5·25-s − 0.185·29-s − 0.538·31-s − 1.01·35-s + 1.31·37-s + 0.312·41-s + 1.06·43-s + 1.60·47-s − 3/7·49-s + 0.137·53-s − 0.404·55-s + 0.520·59-s − 0.512·61-s + 1.11·65-s − 0.488·67-s − 0.237·71-s − 1.40·73-s + 0.227·77-s + 0.787·79-s − 1.30·85-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 29 | \( 1 + T \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 - 11 T + p T^{2} \) | 1.47.al |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + 12 T + p T^{2} \) | 1.73.m |
| 79 | \( 1 - 7 T + p T^{2} \) | 1.79.ah |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−16.22563736798900, −15.54567672070313, −14.91692345752494, −14.68965563176667, −14.27877355522018, −13.33116454766691, −12.67830101368552, −12.22418529692256, −11.79713023575118, −11.14184906563007, −10.74400948002317, −10.09336384613216, −9.247172988251697, −8.725270103046246, −8.026853755178290, −7.628766674021540, −7.200357185350422, −6.290626645381982, −5.636052792993286, −4.760060238310377, −4.246012303283676, −3.809616764954720, −2.815357424205494, −2.068461822566204, −0.9932834161824661, 0,
0.9932834161824661, 2.068461822566204, 2.815357424205494, 3.809616764954720, 4.246012303283676, 4.760060238310377, 5.636052792993286, 6.290626645381982, 7.200357185350422, 7.628766674021540, 8.026853755178290, 8.725270103046246, 9.247172988251697, 10.09336384613216, 10.74400948002317, 11.14184906563007, 11.79713023575118, 12.22418529692256, 12.67830101368552, 13.33116454766691, 14.27877355522018, 14.68965563176667, 14.91692345752494, 15.54567672070313, 16.22563736798900
