| L(s) = 1 | − 2-s − 3·3-s + 4-s + 3·6-s − 8-s + 6·9-s − 11-s − 3·12-s + 2·13-s + 16-s + 7·17-s − 6·18-s − 5·19-s + 22-s − 6·23-s + 3·24-s − 2·26-s − 9·27-s + 4·31-s − 32-s + 3·33-s − 7·34-s + 6·36-s + 5·38-s − 6·39-s − 3·41-s + 4·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s + 1/2·4-s + 1.22·6-s − 0.353·8-s + 2·9-s − 0.301·11-s − 0.866·12-s + 0.554·13-s + 1/4·16-s + 1.69·17-s − 1.41·18-s − 1.14·19-s + 0.213·22-s − 1.25·23-s + 0.612·24-s − 0.392·26-s − 1.73·27-s + 0.718·31-s − 0.176·32-s + 0.522·33-s − 1.20·34-s + 36-s + 0.811·38-s − 0.960·39-s − 0.468·41-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 37 | \( 1 \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 - 7 T + p T^{2} \) | 1.89.ah |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−14.50517485798046, −13.94448127027440, −13.06360550570344, −12.83519037027318, −12.18650325457762, −11.72474502854294, −11.53680074298835, −10.76353430038614, −10.37621485668324, −10.06578748111078, −9.637221549543675, −8.661590875602307, −8.343127681460630, −7.484733658537513, −7.348475038835073, −6.361716787942233, −6.066168376007856, −5.821471600488953, −4.966078757891964, −4.533070844073984, −3.751626311528411, −3.083289453846274, −2.044121861575033, −1.430537634638090, −0.7254513513615607, 0,
0.7254513513615607, 1.430537634638090, 2.044121861575033, 3.083289453846274, 3.751626311528411, 4.533070844073984, 4.966078757891964, 5.821471600488953, 6.066168376007856, 6.361716787942233, 7.348475038835073, 7.484733658537513, 8.343127681460630, 8.661590875602307, 9.637221549543675, 10.06578748111078, 10.37621485668324, 10.76353430038614, 11.53680074298835, 11.72474502854294, 12.18650325457762, 12.83519037027318, 13.06360550570344, 13.94448127027440, 14.50517485798046
