| L(s) = 1 | − 3-s + 7-s + 9-s + 3·11-s + 3·17-s + 5·19-s − 21-s − 6·23-s − 5·25-s − 27-s − 3·29-s − 4·31-s − 3·33-s + 4·37-s + 3·41-s − 8·43-s + 9·47-s + 49-s − 3·51-s − 9·53-s − 5·57-s − 6·59-s + 5·61-s + 63-s + 14·67-s + 6·69-s − 6·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.904·11-s + 0.727·17-s + 1.14·19-s − 0.218·21-s − 1.25·23-s − 25-s − 0.192·27-s − 0.557·29-s − 0.718·31-s − 0.522·33-s + 0.657·37-s + 0.468·41-s − 1.21·43-s + 1.31·47-s + 1/7·49-s − 0.420·51-s − 1.23·53-s − 0.662·57-s − 0.781·59-s + 0.640·61-s + 0.125·63-s + 1.71·67-s + 0.722·69-s − 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−14.56937322819774, −14.12837141245478, −13.70305110322723, −13.11682623968057, −12.35304293067107, −12.11193920509438, −11.53991889557821, −11.24549701904346, −10.58659505849453, −9.922980534162669, −9.543094442452168, −9.152890927367286, −8.207716181109595, −7.859786403671520, −7.321073302940205, −6.709889544544721, −6.039905242558579, −5.630125956318055, −5.133339580736637, −4.326318376359011, −3.832971925693202, −3.308736001135897, −2.291388357810124, −1.607710189595918, −1.004278336808561, 0,
1.004278336808561, 1.607710189595918, 2.291388357810124, 3.308736001135897, 3.832971925693202, 4.326318376359011, 5.133339580736637, 5.630125956318055, 6.039905242558579, 6.709889544544721, 7.321073302940205, 7.859786403671520, 8.207716181109595, 9.152890927367286, 9.543094442452168, 9.922980534162669, 10.58659505849453, 11.24549701904346, 11.53991889557821, 12.11193920509438, 12.35304293067107, 13.11682623968057, 13.70305110322723, 14.12837141245478, 14.56937322819774
