| L(s) = 1 | − 3-s − 2·9-s − 11-s + 5·13-s − 6·17-s − 2·19-s − 6·23-s − 5·25-s + 5·27-s − 3·29-s + 8·31-s + 33-s − 2·37-s − 5·39-s + 6·41-s − 4·43-s + 6·47-s + 6·51-s + 12·53-s + 2·57-s + 3·59-s − 7·61-s − 13·67-s + 6·69-s + 12·71-s + 10·73-s + 5·75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2/3·9-s − 0.301·11-s + 1.38·13-s − 1.45·17-s − 0.458·19-s − 1.25·23-s − 25-s + 0.962·27-s − 0.557·29-s + 1.43·31-s + 0.174·33-s − 0.328·37-s − 0.800·39-s + 0.937·41-s − 0.609·43-s + 0.875·47-s + 0.840·51-s + 1.64·53-s + 0.264·57-s + 0.390·59-s − 0.896·61-s − 1.58·67-s + 0.722·69-s + 1.42·71-s + 1.17·73-s + 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 34496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 13 T + p T^{2} \) | 1.97.an |
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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
−15.38724124372848, −14.78770159886908, −13.94337774658799, −13.63902672033866, −13.31440150976934, −12.50207864524100, −11.97447787823821, −11.49691836845296, −11.02388467570366, −10.58558888010850, −10.05179211066238, −9.255810997901632, −8.702134623721104, −8.294902164580099, −7.747500565144611, −6.845899913275679, −6.341993496321173, −5.885653756982321, −5.460884011491613, −4.480391650093529, −4.119358271195546, −3.363437308341866, −2.456373974972958, −1.931018867645466, −0.8259885723618745, 0,
0.8259885723618745, 1.931018867645466, 2.456373974972958, 3.363437308341866, 4.119358271195546, 4.480391650093529, 5.460884011491613, 5.885653756982321, 6.341993496321173, 6.845899913275679, 7.747500565144611, 8.294902164580099, 8.702134623721104, 9.255810997901632, 10.05179211066238, 10.58558888010850, 11.02388467570366, 11.49691836845296, 11.97447787823821, 12.50207864524100, 13.31440150976934, 13.63902672033866, 13.94337774658799, 14.78770159886908, 15.38724124372848
