| L(s) = 1 | − 3·5-s − 7-s + 4·11-s − 5·17-s − 6·19-s − 6·23-s + 4·25-s − 4·29-s + 3·35-s + 3·37-s − 12·41-s − 3·43-s − 7·47-s − 6·49-s − 2·53-s − 12·55-s − 2·59-s + 12·61-s − 4·67-s + 11·71-s + 6·73-s − 4·77-s + 6·79-s + 10·83-s + 15·85-s − 6·89-s + 18·95-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 0.377·7-s + 1.20·11-s − 1.21·17-s − 1.37·19-s − 1.25·23-s + 4/5·25-s − 0.742·29-s + 0.507·35-s + 0.493·37-s − 1.87·41-s − 0.457·43-s − 1.02·47-s − 6/7·49-s − 0.274·53-s − 1.61·55-s − 0.260·59-s + 1.53·61-s − 0.488·67-s + 1.30·71-s + 0.702·73-s − 0.455·77-s + 0.675·79-s + 1.09·83-s + 1.62·85-s − 0.635·89-s + 1.84·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T + p T^{2} \) | 1.7.b |
| 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.g |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 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.ad |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + 3 T + p T^{2} \) | 1.43.d |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 11 T + p T^{2} \) | 1.71.al |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 - 10 T + p T^{2} \) | 1.83.ak |
| 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
−14.01846781945079, −13.53208905001279, −12.93983371789555, −12.52964018350456, −12.04402429690705, −11.45224939639886, −11.30694627245268, −10.76022040645729, −9.901159443905931, −9.735245564872691, −8.826666451263468, −8.577915045895889, −8.115888360471083, −7.514435483412981, −6.887713439339759, −6.371365423685144, −6.253440135540953, −5.148586169511686, −4.581254645556153, −4.092368303740146, −3.649972375822435, −3.241105754206382, −2.087029128541003, −1.835830911649488, −0.6026655626248948, 0,
0.6026655626248948, 1.835830911649488, 2.087029128541003, 3.241105754206382, 3.649972375822435, 4.092368303740146, 4.581254645556153, 5.148586169511686, 6.253440135540953, 6.371365423685144, 6.887713439339759, 7.514435483412981, 8.115888360471083, 8.577915045895889, 8.826666451263468, 9.735245564872691, 9.901159443905931, 10.76022040645729, 11.30694627245268, 11.45224939639886, 12.04402429690705, 12.52964018350456, 12.93983371789555, 13.53208905001279, 14.01846781945079
