| L(s) = 1 | − 3-s − 3·5-s + 7-s − 2·9-s + 3·15-s + 6·17-s − 4·19-s − 21-s − 3·23-s + 4·25-s + 5·27-s + 6·29-s − 10·31-s − 3·35-s − 8·37-s − 8·43-s + 6·45-s + 6·47-s + 49-s − 6·51-s + 12·53-s + 4·57-s + 3·59-s + 11·61-s − 2·63-s + 2·67-s + 3·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s + 0.377·7-s − 2/3·9-s + 0.774·15-s + 1.45·17-s − 0.917·19-s − 0.218·21-s − 0.625·23-s + 4/5·25-s + 0.962·27-s + 1.11·29-s − 1.79·31-s − 0.507·35-s − 1.31·37-s − 1.21·43-s + 0.894·45-s + 0.875·47-s + 1/7·49-s − 0.840·51-s + 1.64·53-s + 0.529·57-s + 0.390·59-s + 1.40·61-s − 0.251·63-s + 0.244·67-s + 0.361·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 - 11 T + p T^{2} \) | 1.61.al |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−16.11765849027433, −15.47666420165113, −14.91801768974690, −14.43892784571984, −14.03351017205359, −13.14642780305461, −12.46613271278135, −11.90711161404209, −11.83225060992882, −11.11093240479602, −10.49727565113466, −10.17066491773366, −9.124804384317065, −8.419617504100868, −8.238838076205987, −7.444550179410317, −6.956318257072452, −6.186312940315737, −5.382897569099135, −5.115582738087140, −4.062176311767291, −3.720529833746135, −2.916822744878718, −1.931382304106642, −0.8221959911900580, 0,
0.8221959911900580, 1.931382304106642, 2.916822744878718, 3.720529833746135, 4.062176311767291, 5.115582738087140, 5.382897569099135, 6.186312940315737, 6.956318257072452, 7.444550179410317, 8.238838076205987, 8.419617504100868, 9.124804384317065, 10.17066491773366, 10.49727565113466, 11.11093240479602, 11.83225060992882, 11.90711161404209, 12.46613271278135, 13.14642780305461, 14.03351017205359, 14.43892784571984, 14.91801768974690, 15.47666420165113, 16.11765849027433
