| L(s) = 1 | − 2-s − 4-s + 5-s + 3·8-s − 10-s + 6·11-s − 16-s − 4·17-s − 4·19-s − 20-s − 6·22-s + 25-s + 4·29-s − 5·32-s + 4·34-s − 8·37-s + 4·38-s + 3·40-s + 6·41-s + 2·43-s − 6·44-s − 2·47-s − 50-s − 6·53-s + 6·55-s − 4·58-s + 8·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.447·5-s + 1.06·8-s − 0.316·10-s + 1.80·11-s − 1/4·16-s − 0.970·17-s − 0.917·19-s − 0.223·20-s − 1.27·22-s + 1/5·25-s + 0.742·29-s − 0.883·32-s + 0.685·34-s − 1.31·37-s + 0.648·38-s + 0.474·40-s + 0.937·41-s + 0.304·43-s − 0.904·44-s − 0.291·47-s − 0.141·50-s − 0.824·53-s + 0.809·55-s − 0.525·58-s + 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 372645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 372645 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−12.72507229191594, −12.26673828420990, −11.81957790664271, −11.18690835692489, −10.85952258729654, −10.42461377260514, −9.880140953980049, −9.432018078560222, −9.111565195032904, −8.727031839885643, −8.365830630831560, −7.840983787388383, −7.126461356753585, −6.730227331702308, −6.415763579440404, −5.857361553829834, −5.187705168453206, −4.642336255223089, −4.236659037816804, −3.827773940985381, −3.223182984322066, −2.341839373570623, −1.885635909739961, −1.322500225637885, −0.7688444245122103, 0,
0.7688444245122103, 1.322500225637885, 1.885635909739961, 2.341839373570623, 3.223182984322066, 3.827773940985381, 4.236659037816804, 4.642336255223089, 5.187705168453206, 5.857361553829834, 6.415763579440404, 6.730227331702308, 7.126461356753585, 7.840983787388383, 8.365830630831560, 8.727031839885643, 9.111565195032904, 9.432018078560222, 9.880140953980049, 10.42461377260514, 10.85952258729654, 11.18690835692489, 11.81957790664271, 12.26673828420990, 12.72507229191594
