| L(s) = 1 | + 3-s − 2·4-s + 9-s + 11-s − 2·12-s − 13-s + 4·16-s + 6·17-s − 2·19-s − 8·23-s + 27-s + 8·29-s − 4·31-s + 33-s − 2·36-s + 7·37-s − 39-s + 10·41-s − 4·43-s − 2·44-s + 4·48-s + 6·51-s + 2·52-s + 11·53-s − 2·57-s + 59-s − 7·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 1/3·9-s + 0.301·11-s − 0.577·12-s − 0.277·13-s + 16-s + 1.45·17-s − 0.458·19-s − 1.66·23-s + 0.192·27-s + 1.48·29-s − 0.718·31-s + 0.174·33-s − 1/3·36-s + 1.15·37-s − 0.160·39-s + 1.56·41-s − 0.609·43-s − 0.301·44-s + 0.577·48-s + 0.840·51-s + 0.277·52-s + 1.51·53-s − 0.264·57-s + 0.130·59-s − 0.896·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.289916459\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.289916459\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 11 T + p T^{2} \) | 1.53.al |
| 59 | \( 1 - T + p T^{2} \) | 1.59.ab |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 + 7 T + p T^{2} \) | 1.67.h |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 + 5 T + p T^{2} \) | 1.79.f |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 - 5 T + p T^{2} \) | 1.89.af |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−14.55508251634649, −14.04104391901549, −13.74833291837689, −13.00797688656734, −12.65045174704773, −12.01091943718415, −11.78084946163833, −10.73661009662388, −10.18420062883060, −9.931253359346835, −9.300808099204987, −8.851516295225703, −8.233123915078437, −7.802368346941735, −7.412617509314787, −6.463847037693252, −5.886479398018845, −5.421096450719286, −4.505073677951132, −4.245143596510670, −3.559426939131625, −2.951237589288356, −2.174547418308837, −1.293092083118210, −0.5686071965052772,
0.5686071965052772, 1.293092083118210, 2.174547418308837, 2.951237589288356, 3.559426939131625, 4.245143596510670, 4.505073677951132, 5.421096450719286, 5.886479398018845, 6.463847037693252, 7.412617509314787, 7.802368346941735, 8.233123915078437, 8.851516295225703, 9.300808099204987, 9.931253359346835, 10.18420062883060, 10.73661009662388, 11.78084946163833, 12.01091943718415, 12.65045174704773, 13.00797688656734, 13.74833291837689, 14.04104391901549, 14.55508251634649
