| L(s) = 1 | + 3-s − 2·7-s + 9-s − 2·11-s − 4·13-s − 17-s + 19-s − 2·21-s + 9·23-s + 27-s + 8·31-s − 2·33-s − 3·37-s − 4·39-s + 7·41-s + 4·43-s + 8·47-s − 3·49-s − 51-s − 9·53-s + 57-s + 5·59-s + 7·61-s − 2·63-s + 8·67-s + 9·69-s + 3·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.603·11-s − 1.10·13-s − 0.242·17-s + 0.229·19-s − 0.436·21-s + 1.87·23-s + 0.192·27-s + 1.43·31-s − 0.348·33-s − 0.493·37-s − 0.640·39-s + 1.09·41-s + 0.609·43-s + 1.16·47-s − 3/7·49-s − 0.140·51-s − 1.23·53-s + 0.132·57-s + 0.650·59-s + 0.896·61-s − 0.251·63-s + 0.977·67-s + 1.08·69-s + 0.356·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| 19 | \( 1 - T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 - 7 T + p T^{2} \) | 1.41.ah |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + T + p T^{2} \) | 1.83.b |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−12.66313818589880, −12.43875922376331, −11.85336872708035, −11.26745806771142, −10.88643638608320, −10.24883075023430, −10.02919628895317, −9.460533125854096, −9.091604017500994, −8.752012390224236, −8.025128006618699, −7.682232818425429, −7.256937323266066, −6.649526803213057, −6.482472710049731, −5.634375379897459, −5.147496020993622, −4.790669647550352, −4.174903827327857, −3.626372202540422, −2.960848144264898, −2.626452807160832, −2.337918244695941, −1.329272483699915, −0.7762025623905202, 0,
0.7762025623905202, 1.329272483699915, 2.337918244695941, 2.626452807160832, 2.960848144264898, 3.626372202540422, 4.174903827327857, 4.790669647550352, 5.147496020993622, 5.634375379897459, 6.482472710049731, 6.649526803213057, 7.256937323266066, 7.682232818425429, 8.025128006618699, 8.752012390224236, 9.091604017500994, 9.460533125854096, 10.02919628895317, 10.24883075023430, 10.88643638608320, 11.26745806771142, 11.85336872708035, 12.43875922376331, 12.66313818589880
