| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s + 11-s + 12-s + 6·13-s + 16-s + 6·17-s + 18-s + 2·19-s + 22-s + 24-s + 6·26-s + 27-s + 8·29-s + 32-s + 33-s + 6·34-s + 36-s + 10·37-s + 2·38-s + 6·39-s − 8·41-s + 44-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 + 0.301·11-s + 0.288·12-s + 1.66·13-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.458·19-s + 0.213·22-s + 0.204·24-s + 1.17·26-s + 0.192·27-s + 1.48·29-s + 0.176·32-s + 0.174·33-s + 1.02·34-s + 1/6·36-s + 1.64·37-s + 0.324·38-s + 0.960·39-s − 1.24·41-s + 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.128609609\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.128609609\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−14.08171928997491, −13.50326468053826, −13.13246131452454, −12.61775111396557, −11.95364194881677, −11.68104960703230, −11.08911850603687, −10.47008815865946, −10.03498510331821, −9.523849234991962, −8.821846666758944, −8.380458322685307, −7.853443899149057, −7.440352929937061, −6.588367193535587, −6.254315548005563, −5.769814382484894, −4.987366278603826, −4.560537165950010, −3.729142611918841, −3.409567465424237, −2.950922649305151, −2.100530628744036, −1.285089060433142, −0.9054223760076032,
0.9054223760076032, 1.285089060433142, 2.100530628744036, 2.950922649305151, 3.409567465424237, 3.729142611918841, 4.560537165950010, 4.987366278603826, 5.769814382484894, 6.254315548005563, 6.588367193535587, 7.440352929937061, 7.853443899149057, 8.380458322685307, 8.821846666758944, 9.523849234991962, 10.03498510331821, 10.47008815865946, 11.08911850603687, 11.68104960703230, 11.95364194881677, 12.61775111396557, 13.13246131452454, 13.50326468053826, 14.08171928997491
