| L(s) = 1 | + 2·5-s − 2·7-s + 2·13-s − 2·17-s − 4·19-s − 6·23-s − 25-s + 2·29-s − 4·35-s − 2·37-s + 6·41-s + 8·43-s − 2·47-s − 3·49-s + 10·53-s + 4·59-s + 14·61-s + 4·65-s + 4·67-s − 6·71-s + 8·73-s + 2·79-s + 12·83-s − 4·85-s − 8·89-s − 4·91-s − 8·95-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.755·7-s + 0.554·13-s − 0.485·17-s − 0.917·19-s − 1.25·23-s − 1/5·25-s + 0.371·29-s − 0.676·35-s − 0.328·37-s + 0.937·41-s + 1.21·43-s − 0.291·47-s − 3/7·49-s + 1.37·53-s + 0.520·59-s + 1.79·61-s + 0.496·65-s + 0.488·67-s − 0.712·71-s + 0.936·73-s + 0.225·79-s + 1.31·83-s − 0.433·85-s − 0.847·89-s − 0.419·91-s − 0.820·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17424 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17424 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 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
−16.04634715817292, −15.78057420982180, −14.98070884688639, −14.39941994815040, −13.85482985003301, −13.35515314286933, −12.92751330508770, −12.35725116992071, −11.72281894599968, −10.96900284508357, −10.49228209915687, −9.856727002732818, −9.500183713280355, −8.764253750808926, −8.288340348981413, −7.503699979463203, −6.654615045497582, −6.294285390964914, −5.768118078461639, −5.083139853085300, −4.040299765356000, −3.762476082050356, −2.521346712570419, −2.223285709251439, −1.148309353306660, 0,
1.148309353306660, 2.223285709251439, 2.521346712570419, 3.762476082050356, 4.040299765356000, 5.083139853085300, 5.768118078461639, 6.294285390964914, 6.654615045497582, 7.503699979463203, 8.288340348981413, 8.764253750808926, 9.500183713280355, 9.856727002732818, 10.49228209915687, 10.96900284508357, 11.72281894599968, 12.35725116992071, 12.92751330508770, 13.35515314286933, 13.85482985003301, 14.39941994815040, 14.98070884688639, 15.78057420982180, 16.04634715817292
