| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s − 4·11-s + 12-s − 13-s + 16-s − 4·17-s − 18-s + 5·19-s + 4·22-s − 4·23-s − 24-s + 26-s + 27-s + 7·29-s − 32-s − 4·33-s + 4·34-s + 36-s − 3·37-s − 5·38-s − 39-s − 9·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 1.20·11-s + 0.288·12-s − 0.277·13-s + 1/4·16-s − 0.970·17-s − 0.235·18-s + 1.14·19-s + 0.852·22-s − 0.834·23-s − 0.204·24-s + 0.196·26-s + 0.192·27-s + 1.29·29-s − 0.176·32-s − 0.696·33-s + 0.685·34-s + 1/6·36-s − 0.493·37-s − 0.811·38-s − 0.160·39-s − 1.40·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 7 T + p T^{2} \) | 1.29.ah |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + 5 T + p T^{2} \) | 1.53.f |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 9 T + p T^{2} \) | 1.67.aj |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 + 5 T + p T^{2} \) | 1.79.f |
| 83 | \( 1 + 18 T + p T^{2} \) | 1.83.s |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−14.17852655634150, −13.53290184775715, −13.18547544699984, −12.47407518323092, −12.14156516887632, −11.49836073993062, −11.03212629371710, −10.43406866757054, −9.939344339303888, −9.765509590387656, −8.998263473465667, −8.442974588743415, −8.204051452337708, −7.620144921495737, −7.057438854462017, −6.700141903690515, −5.924480556658157, −5.327401192020320, −4.777562067579391, −4.179434302680369, −3.247006532421568, −2.968769152889545, −2.201702289569964, −1.765386715090937, −0.7969743664061135, 0,
0.7969743664061135, 1.765386715090937, 2.201702289569964, 2.968769152889545, 3.247006532421568, 4.179434302680369, 4.777562067579391, 5.327401192020320, 5.924480556658157, 6.700141903690515, 7.057438854462017, 7.620144921495737, 8.204051452337708, 8.442974588743415, 8.998263473465667, 9.765509590387656, 9.939344339303888, 10.43406866757054, 11.03212629371710, 11.49836073993062, 12.14156516887632, 12.47407518323092, 13.18547544699984, 13.53290184775715, 14.17852655634150
