| L(s) = 1 | + 3-s − 5-s + 3·7-s + 9-s + 3·11-s − 15-s − 17-s + 6·19-s + 3·21-s − 5·23-s + 25-s + 27-s − 6·29-s − 2·31-s + 3·33-s − 3·35-s − 7·37-s − 3·41-s − 8·43-s − 45-s + 2·47-s + 2·49-s − 51-s − 53-s − 3·55-s + 6·57-s − 15·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.13·7-s + 1/3·9-s + 0.904·11-s − 0.258·15-s − 0.242·17-s + 1.37·19-s + 0.654·21-s − 1.04·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.359·31-s + 0.522·33-s − 0.507·35-s − 1.15·37-s − 0.468·41-s − 1.21·43-s − 0.149·45-s + 0.291·47-s + 2/7·49-s − 0.140·51-s − 0.137·53-s − 0.404·55-s + 0.794·57-s − 1.92·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 5 T + p T^{2} \) | 1.23.f |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 15 T + p T^{2} \) | 1.61.p |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 13 T + p T^{2} \) | 1.79.n |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + T + p T^{2} \) | 1.89.b |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
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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
−15.76960985422387, −15.28442135173425, −14.81349683718460, −14.27449208364779, −13.85251074158379, −13.41758550855021, −12.52620081743502, −11.94294407170935, −11.63577320318303, −11.05362807407571, −10.39471304840644, −9.676320144627584, −9.186645510114120, −8.514290226332103, −8.131524092355225, −7.345762818360174, −7.153878018374171, −6.153450329581632, −5.451027168715234, −4.777894839737492, −4.131020906456651, −3.551987695407266, −2.855777716545377, −1.625436083000903, −1.530669581869337, 0,
1.530669581869337, 1.625436083000903, 2.855777716545377, 3.551987695407266, 4.131020906456651, 4.777894839737492, 5.451027168715234, 6.153450329581632, 7.153878018374171, 7.345762818360174, 8.131524092355225, 8.514290226332103, 9.186645510114120, 9.676320144627584, 10.39471304840644, 11.05362807407571, 11.63577320318303, 11.94294407170935, 12.52620081743502, 13.41758550855021, 13.85251074158379, 14.27449208364779, 14.81349683718460, 15.28442135173425, 15.76960985422387
