| L(s) = 1 | − 3-s + 7-s + 9-s − 6·11-s + 13-s + 3·17-s + 4·19-s − 21-s − 3·23-s − 27-s − 3·29-s + 5·31-s + 6·33-s + 10·37-s − 39-s + 9·41-s + 43-s + 49-s − 3·51-s − 9·53-s − 4·57-s − 9·59-s − 11·61-s + 63-s + 4·67-s + 3·69-s − 12·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.80·11-s + 0.277·13-s + 0.727·17-s + 0.917·19-s − 0.218·21-s − 0.625·23-s − 0.192·27-s − 0.557·29-s + 0.898·31-s + 1.04·33-s + 1.64·37-s − 0.160·39-s + 1.40·41-s + 0.152·43-s + 1/7·49-s − 0.420·51-s − 1.23·53-s − 0.529·57-s − 1.17·59-s − 1.40·61-s + 0.125·63-s + 0.488·67-s + 0.361·69-s − 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| good | 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 + 11 T + p T^{2} \) | 1.61.l |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−15.47698344687308, −14.67590688763980, −14.23423340348809, −13.64190636345629, −12.93198775839519, −12.86211380493644, −11.91143681209507, −11.68390120611431, −10.81080923440402, −10.71950920115585, −9.884720670352183, −9.611663257245748, −8.761894036036676, −7.974274529663948, −7.663658213707019, −7.338974279182299, −6.165614081605839, −5.900415578787894, −5.329543785694607, −4.651671635754719, −4.215518703645140, −3.083313552148040, −2.757992353686380, −1.742241674805126, −0.9467410604047762, 0,
0.9467410604047762, 1.742241674805126, 2.757992353686380, 3.083313552148040, 4.215518703645140, 4.651671635754719, 5.329543785694607, 5.900415578787894, 6.165614081605839, 7.338974279182299, 7.663658213707019, 7.974274529663948, 8.761894036036676, 9.611663257245748, 9.884720670352183, 10.71950920115585, 10.81080923440402, 11.68390120611431, 11.91143681209507, 12.86211380493644, 12.93198775839519, 13.64190636345629, 14.23423340348809, 14.67590688763980, 15.47698344687308
