| L(s) = 1 | − 2-s + 4-s − 7-s − 8-s + 11-s + 13-s + 14-s + 16-s − 17-s − 4·19-s − 22-s − 8·23-s − 26-s − 28-s + 8·29-s − 32-s + 34-s − 7·37-s + 4·38-s + 8·41-s − 11·43-s + 44-s + 8·46-s − 47-s − 6·49-s + 52-s + 2·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 0.301·11-s + 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.242·17-s − 0.917·19-s − 0.213·22-s − 1.66·23-s − 0.196·26-s − 0.188·28-s + 1.48·29-s − 0.176·32-s + 0.171·34-s − 1.15·37-s + 0.648·38-s + 1.24·41-s − 1.67·43-s + 0.150·44-s + 1.17·46-s − 0.145·47-s − 6/7·49-s + 0.138·52-s + 0.274·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5122083492\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5122083492\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 + 11 T + p T^{2} \) | 1.43.l |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 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
−14.22515928934373, −13.69743193899020, −13.34256827367924, −12.48007936225048, −12.19848079512232, −11.79019919148981, −11.03616429073975, −10.59850221173978, −10.16769647113140, −9.675356758223257, −9.043345415369789, −8.663726461198447, −7.992922138581272, −7.762719875299828, −6.734555399715045, −6.549013127871183, −6.046220883628081, −5.346276583202277, −4.484955508906177, −4.088587166604232, −3.241897845569760, −2.738204300541557, −1.855156079023228, −1.413209931671082, −0.2660740739984768,
0.2660740739984768, 1.413209931671082, 1.855156079023228, 2.738204300541557, 3.241897845569760, 4.088587166604232, 4.484955508906177, 5.346276583202277, 6.046220883628081, 6.549013127871183, 6.734555399715045, 7.762719875299828, 7.992922138581272, 8.663726461198447, 9.043345415369789, 9.675356758223257, 10.16769647113140, 10.59850221173978, 11.03616429073975, 11.79019919148981, 12.19848079512232, 12.48007936225048, 13.34256827367924, 13.69743193899020, 14.22515928934373
