| L(s) = 1 | − 3-s + 4·7-s + 9-s + 13-s + 2·17-s + 4·19-s − 4·21-s + 2·23-s − 27-s + 6·29-s − 10·31-s − 2·37-s − 39-s + 6·41-s − 4·43-s + 9·49-s − 2·51-s − 6·53-s − 4·57-s − 4·59-s − 6·61-s + 4·63-s − 2·67-s − 2·69-s − 10·71-s − 10·73-s − 4·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.51·7-s + 1/3·9-s + 0.277·13-s + 0.485·17-s + 0.917·19-s − 0.872·21-s + 0.417·23-s − 0.192·27-s + 1.11·29-s − 1.79·31-s − 0.328·37-s − 0.160·39-s + 0.937·41-s − 0.609·43-s + 9/7·49-s − 0.280·51-s − 0.824·53-s − 0.529·57-s − 0.520·59-s − 0.768·61-s + 0.503·63-s − 0.244·67-s − 0.240·69-s − 1.18·71-s − 1.17·73-s − 0.450·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−13.91406071716298, −13.19084677613675, −12.81186307798791, −12.14517475393759, −11.81804936057253, −11.36006524549426, −10.89890391748575, −10.58441201741282, −9.953890757033975, −9.389052245389017, −8.828566420393921, −8.363322906027722, −7.788510225250799, −7.326006031984882, −7.005750944554653, −6.076991351354496, −5.718051981207303, −5.216306665060060, −4.636565365317488, −4.353162666839978, −3.443516444747208, −2.981536015532261, −2.038299958069531, −1.439960839965743, −1.048106362170043, 0,
1.048106362170043, 1.439960839965743, 2.038299958069531, 2.981536015532261, 3.443516444747208, 4.353162666839978, 4.636565365317488, 5.216306665060060, 5.718051981207303, 6.076991351354496, 7.005750944554653, 7.326006031984882, 7.788510225250799, 8.363322906027722, 8.828566420393921, 9.389052245389017, 9.953890757033975, 10.58441201741282, 10.89890391748575, 11.36006524549426, 11.81804936057253, 12.14517475393759, 12.81186307798791, 13.19084677613675, 13.91406071716298
