| L(s) = 1 | − 2·2-s − 3-s + 2·4-s + 2·6-s − 7-s + 9-s − 3·11-s − 2·12-s + 2·14-s − 4·16-s + 3·17-s − 2·18-s − 2·19-s + 21-s + 6·22-s − 27-s − 2·28-s − 3·29-s − 4·31-s + 8·32-s + 3·33-s − 6·34-s + 2·36-s + 4·37-s + 4·38-s − 4·41-s − 2·42-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s + 0.816·6-s − 0.377·7-s + 1/3·9-s − 0.904·11-s − 0.577·12-s + 0.534·14-s − 16-s + 0.727·17-s − 0.471·18-s − 0.458·19-s + 0.218·21-s + 1.27·22-s − 0.192·27-s − 0.377·28-s − 0.557·29-s − 0.718·31-s + 1.41·32-s + 0.522·33-s − 1.02·34-s + 1/3·36-s + 0.657·37-s + 0.648·38-s − 0.624·41-s − 0.308·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 2 | \( 1 + p T + p T^{2} \) | 1.2.c |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 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 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 11 T + p T^{2} \) | 1.47.l |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−14.44929680815614, −13.85058567830965, −13.14668233000718, −12.90618899597344, −12.40540292979280, −11.63118251997330, −11.24073850152208, −10.85299334408262, −10.23048849756098, −9.927924249681665, −9.511981424030963, −8.936423422467868, −8.278607144594006, −7.913184340273138, −7.486617061967339, −6.753012459401623, −6.509514993733240, −5.635031887581048, −5.253020344191298, −4.580896852489140, −3.888860285388413, −3.131858400593451, −2.455632969974905, −1.688474563514403, −1.148559104992144, 0, 0,
1.148559104992144, 1.688474563514403, 2.455632969974905, 3.131858400593451, 3.888860285388413, 4.580896852489140, 5.253020344191298, 5.635031887581048, 6.509514993733240, 6.753012459401623, 7.486617061967339, 7.913184340273138, 8.278607144594006, 8.936423422467868, 9.511981424030963, 9.927924249681665, 10.23048849756098, 10.85299334408262, 11.24073850152208, 11.63118251997330, 12.40540292979280, 12.90618899597344, 13.14668233000718, 13.85058567830965, 14.44929680815614
