| L(s) = 1 | + 3-s + 3·5-s + 3·7-s + 9-s + 3·11-s + 3·15-s + 17-s + 19-s + 3·21-s + 4·25-s + 27-s − 8·29-s − 2·31-s + 3·33-s + 9·35-s − 4·37-s − 12·41-s − 43-s + 3·45-s + 9·47-s + 2·49-s + 51-s + 6·53-s + 9·55-s + 57-s − 6·59-s − 61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s + 1.13·7-s + 1/3·9-s + 0.904·11-s + 0.774·15-s + 0.242·17-s + 0.229·19-s + 0.654·21-s + 4/5·25-s + 0.192·27-s − 1.48·29-s − 0.359·31-s + 0.522·33-s + 1.52·35-s − 0.657·37-s − 1.87·41-s − 0.152·43-s + 0.447·45-s + 1.31·47-s + 2/7·49-s + 0.140·51-s + 0.824·53-s + 1.21·55-s + 0.132·57-s − 0.781·59-s − 0.128·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1824 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1824 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.279797776\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.279797776\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 19 | \( 1 - T \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−9.101238015196153235891929103797, −8.739742529675211537680832000732, −7.68480034566561980728370681208, −6.97490045236361700205558801297, −5.93913106016740764907159143329, −5.29110799616034856454488976968, −4.31976547466127888889844461893, −3.26094354372819570090498374626, −1.97176209830415245680106247251, −1.49634774917676401147922867412,
1.49634774917676401147922867412, 1.97176209830415245680106247251, 3.26094354372819570090498374626, 4.31976547466127888889844461893, 5.29110799616034856454488976968, 5.93913106016740764907159143329, 6.97490045236361700205558801297, 7.68480034566561980728370681208, 8.739742529675211537680832000732, 9.101238015196153235891929103797
