| L(s) = 1 | + 3-s + 5-s − 4·7-s + 9-s + 11-s − 6·13-s + 15-s − 2·17-s − 4·19-s − 4·21-s − 8·23-s + 25-s + 27-s + 6·29-s + 33-s − 4·35-s − 2·37-s − 6·39-s + 2·41-s − 8·43-s + 45-s + 9·49-s − 2·51-s − 2·53-s + 55-s − 4·57-s + 4·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s + 0.301·11-s − 1.66·13-s + 0.258·15-s − 0.485·17-s − 0.917·19-s − 0.872·21-s − 1.66·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.174·33-s − 0.676·35-s − 0.328·37-s − 0.960·39-s + 0.312·41-s − 1.21·43-s + 0.149·45-s + 9/7·49-s − 0.280·51-s − 0.274·53-s + 0.134·55-s − 0.529·57-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 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
−9.333730813049933185735496650902, −8.567403487206224933564651554616, −7.56237980873441436401389019686, −6.65367439925152600648731332061, −6.17793908705338672919867029004, −4.88542583910507496729378372815, −3.92144876445360075975332781553, −2.85919212079310983846989540877, −2.07891804918662471445559880221, 0,
2.07891804918662471445559880221, 2.85919212079310983846989540877, 3.92144876445360075975332781553, 4.88542583910507496729378372815, 6.17793908705338672919867029004, 6.65367439925152600648731332061, 7.56237980873441436401389019686, 8.567403487206224933564651554616, 9.333730813049933185735496650902
