| L(s) = 1 | + 2-s − 2·3-s + 4-s − 2·6-s − 7-s + 8-s + 9-s + 3·11-s − 2·12-s + 13-s − 14-s + 16-s + 18-s − 4·19-s + 2·21-s + 3·22-s + 3·23-s − 2·24-s + 26-s + 4·27-s − 28-s + 6·29-s + 10·31-s + 32-s − 6·33-s + 36-s − 10·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.904·11-s − 0.577·12-s + 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.235·18-s − 0.917·19-s + 0.436·21-s + 0.639·22-s + 0.625·23-s − 0.408·24-s + 0.196·26-s + 0.769·27-s − 0.188·28-s + 1.11·29-s + 1.79·31-s + 0.176·32-s − 1.04·33-s + 1/6·36-s − 1.64·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.124584870\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.124584870\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−16.24798846677944, −15.44046945997918, −15.18385008665943, −14.22939547698331, −13.98891607692805, −13.21002309210557, −12.63076736324287, −12.11697681775452, −11.70254819389354, −11.20318251143953, −10.42367345494632, −10.24235561914300, −9.204469594175560, −8.627496664373348, −7.920374578952103, −6.866738844374864, −6.510795094265479, −6.200204586786368, −5.365677106811577, −4.757899561332664, −4.210968514675879, −3.354589884810430, −2.635685562024001, −1.512179711033136, −0.6377487421943353,
0.6377487421943353, 1.512179711033136, 2.635685562024001, 3.354589884810430, 4.210968514675879, 4.757899561332664, 5.365677106811577, 6.200204586786368, 6.510795094265479, 6.866738844374864, 7.920374578952103, 8.627496664373348, 9.204469594175560, 10.24235561914300, 10.42367345494632, 11.20318251143953, 11.70254819389354, 12.11697681775452, 12.63076736324287, 13.21002309210557, 13.98891607692805, 14.22939547698331, 15.18385008665943, 15.44046945997918, 16.24798846677944
