| L(s) = 1 | + 5-s + 3·7-s − 3·11-s + 4·13-s − 2·17-s − 6·19-s + 3·23-s − 4·25-s − 2·31-s + 3·35-s + 37-s − 3·41-s − 10·43-s + 7·47-s + 2·49-s + 3·53-s − 3·55-s + 10·59-s + 8·61-s + 4·65-s − 8·67-s − 8·71-s − 7·73-s − 9·77-s − 12·79-s − 11·83-s − 2·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.13·7-s − 0.904·11-s + 1.10·13-s − 0.485·17-s − 1.37·19-s + 0.625·23-s − 4/5·25-s − 0.359·31-s + 0.507·35-s + 0.164·37-s − 0.468·41-s − 1.52·43-s + 1.02·47-s + 2/7·49-s + 0.412·53-s − 0.404·55-s + 1.30·59-s + 1.02·61-s + 0.496·65-s − 0.977·67-s − 0.949·71-s − 0.819·73-s − 1.02·77-s − 1.35·79-s − 1.20·83-s − 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.084122060\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.084122060\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 397 | \( 1 + T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + 11 T + p T^{2} \) | 1.83.l |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−12.98293354203134, −12.70679707060411, −11.81993752527410, −11.41329753124028, −11.20731091747366, −10.52754956410874, −10.28275928158123, −9.828304776766771, −8.876084186586838, −8.752240794326460, −8.308099087684182, −7.855016067380814, −7.201165380537388, −6.782807410825978, −6.158995902469337, −5.595902915873447, −5.363812434927779, −4.579885814642755, −4.269436093282999, −3.659899286507451, −2.893548393955086, −2.339092771752586, −1.749065665293383, −1.371317637674349, −0.3820640618670147,
0.3820640618670147, 1.371317637674349, 1.749065665293383, 2.339092771752586, 2.893548393955086, 3.659899286507451, 4.269436093282999, 4.579885814642755, 5.363812434927779, 5.595902915873447, 6.158995902469337, 6.782807410825978, 7.201165380537388, 7.855016067380814, 8.308099087684182, 8.752240794326460, 8.876084186586838, 9.828304776766771, 10.28275928158123, 10.52754956410874, 11.20731091747366, 11.41329753124028, 11.81993752527410, 12.70679707060411, 12.98293354203134
