| L(s) = 1 | − 3·3-s + 7-s + 6·9-s + 4·11-s − 5·17-s + 6·19-s − 3·21-s + 6·23-s − 9·27-s − 4·29-s − 12·33-s − 3·37-s − 12·41-s + 3·43-s − 7·47-s − 6·49-s + 15·51-s + 2·53-s − 18·57-s − 2·59-s − 12·61-s + 6·63-s − 4·67-s − 18·69-s − 11·71-s + 6·73-s + 4·77-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.377·7-s + 2·9-s + 1.20·11-s − 1.21·17-s + 1.37·19-s − 0.654·21-s + 1.25·23-s − 1.73·27-s − 0.742·29-s − 2.08·33-s − 0.493·37-s − 1.87·41-s + 0.457·43-s − 1.02·47-s − 6/7·49-s + 2.10·51-s + 0.274·53-s − 2.38·57-s − 0.260·59-s − 1.53·61-s + 0.755·63-s − 0.488·67-s − 2.16·69-s − 1.30·71-s + 0.702·73-s + 0.455·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 - 3 T + p T^{2} \) | 1.43.ad |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 11 T + p T^{2} \) | 1.71.l |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 + 10 T + p T^{2} \) | 1.83.k |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−13.66578563587349, −13.05276161682202, −12.63254591947157, −12.08682865899706, −11.61195047365390, −11.31062375256823, −11.11973410136655, −10.40406352629928, −9.936596904179696, −9.419957838931247, −8.875013591561849, −8.452941685405183, −7.406612556314860, −7.236246042307962, −6.690595355039620, −6.224206258638348, −5.742828101562064, −5.127098211143595, −4.716248825572744, −4.364489824769542, −3.506828713318381, −3.041924854481273, −1.753280386472292, −1.544763372097344, −0.7619007547071479, 0,
0.7619007547071479, 1.544763372097344, 1.753280386472292, 3.041924854481273, 3.506828713318381, 4.364489824769542, 4.716248825572744, 5.127098211143595, 5.742828101562064, 6.224206258638348, 6.690595355039620, 7.236246042307962, 7.406612556314860, 8.452941685405183, 8.875013591561849, 9.419957838931247, 9.936596904179696, 10.40406352629928, 11.11973410136655, 11.31062375256823, 11.61195047365390, 12.08682865899706, 12.63254591947157, 13.05276161682202, 13.66578563587349
