| L(s) = 1 | + 3·5-s + 3·7-s + 3·17-s + 6·19-s + 6·23-s + 4·25-s + 9·35-s − 3·37-s − 43-s − 3·47-s + 2·49-s + 6·53-s + 6·59-s − 8·61-s + 12·67-s + 15·71-s − 6·73-s − 10·79-s + 6·83-s + 9·85-s − 6·89-s + 18·95-s − 12·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 1.13·7-s + 0.727·17-s + 1.37·19-s + 1.25·23-s + 4/5·25-s + 1.52·35-s − 0.493·37-s − 0.152·43-s − 0.437·47-s + 2/7·49-s + 0.824·53-s + 0.781·59-s − 1.02·61-s + 1.46·67-s + 1.78·71-s − 0.702·73-s − 1.12·79-s + 0.658·83-s + 0.976·85-s − 0.635·89-s + 1.84·95-s − 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.493295537\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.493295537\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−15.32584402208799, −14.71262408642239, −14.22684203352791, −13.92857391549895, −13.34704945831928, −12.81288431265149, −12.14927511074700, −11.51912138222128, −11.10856366286174, −10.42768955221570, −9.864486508663025, −9.472775546932706, −8.798697017523555, −8.248281461811509, −7.540764509698620, −7.038858984567775, −6.307080416774806, −5.524307775343883, −5.244934232226745, −4.732141669091420, −3.692903488581725, −2.972974101190827, −2.210858456658593, −1.478579201666905, −0.9373336881138292,
0.9373336881138292, 1.478579201666905, 2.210858456658593, 2.972974101190827, 3.692903488581725, 4.732141669091420, 5.244934232226745, 5.524307775343883, 6.307080416774806, 7.038858984567775, 7.540764509698620, 8.248281461811509, 8.798697017523555, 9.472775546932706, 9.864486508663025, 10.42768955221570, 11.10856366286174, 11.51912138222128, 12.14927511074700, 12.81288431265149, 13.34704945831928, 13.92857391549895, 14.22684203352791, 14.71262408642239, 15.32584402208799
