| L(s) = 1 | + 7-s + 3·11-s + 3·13-s + 8·17-s − 7·19-s − 23-s + 7·29-s − 10·31-s − 4·37-s − 11·41-s + 5·43-s − 10·47-s − 6·49-s − 6·53-s + 8·59-s + 8·61-s + 10·71-s − 11·73-s + 3·77-s − 3·79-s − 83-s + 6·89-s + 3·91-s + 14·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 0.904·11-s + 0.832·13-s + 1.94·17-s − 1.60·19-s − 0.208·23-s + 1.29·29-s − 1.79·31-s − 0.657·37-s − 1.71·41-s + 0.762·43-s − 1.45·47-s − 6/7·49-s − 0.824·53-s + 1.04·59-s + 1.02·61-s + 1.18·71-s − 1.28·73-s + 0.341·77-s − 0.337·79-s − 0.109·83-s + 0.635·89-s + 0.314·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 331200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.038655215\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.038655215\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 - 8 T + p T^{2} \) | 1.17.ai |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 29 | \( 1 - 7 T + p T^{2} \) | 1.29.ah |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 11 T + p T^{2} \) | 1.41.l |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 3 T + p T^{2} \) | 1.79.d |
| 83 | \( 1 + T + p T^{2} \) | 1.83.b |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−12.70114346095847, −12.08127354697229, −11.67754358218777, −11.33740402949819, −10.72182000200512, −10.36626970493782, −9.897296865313498, −9.466322973344339, −8.745042379911355, −8.505198680953141, −8.128469296096067, −7.555060229309203, −6.934187754443852, −6.565432822685972, −6.052049696875355, −5.619702523361447, −4.999832904652332, −4.581351203480288, −3.884854246418217, −3.436263992222535, −3.192784459943836, −2.096540329135595, −1.748339033035981, −1.189247751983938, −0.4752558065876469,
0.4752558065876469, 1.189247751983938, 1.748339033035981, 2.096540329135595, 3.192784459943836, 3.436263992222535, 3.884854246418217, 4.581351203480288, 4.999832904652332, 5.619702523361447, 6.052049696875355, 6.565432822685972, 6.934187754443852, 7.555060229309203, 8.128469296096067, 8.505198680953141, 8.745042379911355, 9.466322973344339, 9.897296865313498, 10.36626970493782, 10.72182000200512, 11.33740402949819, 11.67754358218777, 12.08127354697229, 12.70114346095847
