| L(s) = 1 | − 3-s + 5-s + 7-s − 2·9-s − 3·11-s − 15-s − 3·17-s + 7·19-s − 21-s + 3·23-s + 25-s + 5·27-s + 3·29-s + 4·31-s + 3·33-s + 35-s − 7·37-s − 9·41-s − 11·43-s − 2·45-s − 6·49-s + 3·51-s − 6·53-s − 3·55-s − 7·57-s + 3·59-s + 11·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s − 2/3·9-s − 0.904·11-s − 0.258·15-s − 0.727·17-s + 1.60·19-s − 0.218·21-s + 0.625·23-s + 1/5·25-s + 0.962·27-s + 0.557·29-s + 0.718·31-s + 0.522·33-s + 0.169·35-s − 1.15·37-s − 1.40·41-s − 1.67·43-s − 0.298·45-s − 6/7·49-s + 0.420·51-s − 0.824·53-s − 0.404·55-s − 0.927·57-s + 0.390·59-s + 1.40·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 + 11 T + p T^{2} \) | 1.43.l |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 - 11 T + p T^{2} \) | 1.61.al |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−16.36294847207808, −16.09193812689933, −15.37472143482233, −14.86987085264087, −14.09691926451689, −13.67118614464041, −13.24135260543230, −12.44450830936354, −11.84330411795959, −11.38173109667984, −10.88451906088887, −10.15080268907504, −9.787083598457743, −8.877144073857117, −8.362324489001090, −7.824030470227425, −6.811775999698847, −6.577587417740347, −5.562102638385271, −5.080690922383921, −4.848353418673216, −3.487521383705842, −2.926435921812421, −2.066557621072841, −1.085326756946610, 0,
1.085326756946610, 2.066557621072841, 2.926435921812421, 3.487521383705842, 4.848353418673216, 5.080690922383921, 5.562102638385271, 6.577587417740347, 6.811775999698847, 7.824030470227425, 8.362324489001090, 8.877144073857117, 9.787083598457743, 10.15080268907504, 10.88451906088887, 11.38173109667984, 11.84330411795959, 12.44450830936354, 13.24135260543230, 13.67118614464041, 14.09691926451689, 14.86987085264087, 15.37472143482233, 16.09193812689933, 16.36294847207808
