| L(s) = 1 | + 2·5-s − 7-s + 3·13-s + 6·17-s − 4·19-s + 23-s − 25-s + 3·31-s − 2·35-s − 3·37-s + 11·41-s + 2·43-s + 6·47-s − 6·49-s + 6·53-s − 7·59-s − 10·61-s + 6·65-s − 3·67-s + 8·71-s − 2·73-s + 4·79-s + 15·83-s + 12·85-s − 6·89-s − 3·91-s − 8·95-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.377·7-s + 0.832·13-s + 1.45·17-s − 0.917·19-s + 0.208·23-s − 1/5·25-s + 0.538·31-s − 0.338·35-s − 0.493·37-s + 1.71·41-s + 0.304·43-s + 0.875·47-s − 6/7·49-s + 0.824·53-s − 0.911·59-s − 1.28·61-s + 0.744·65-s − 0.366·67-s + 0.949·71-s − 0.234·73-s + 0.450·79-s + 1.64·83-s + 1.30·85-s − 0.635·89-s − 0.314·91-s − 0.820·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.444183885\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.444183885\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 - 11 T + p T^{2} \) | 1.41.al |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 7 T + p T^{2} \) | 1.59.h |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 15 T + p T^{2} \) | 1.83.ap |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.11330624362387, −12.61459984818849, −12.16879023318302, −11.80996171762984, −10.94499639741320, −10.72801391946115, −10.28630973765073, −9.712760632118619, −9.302148072086691, −8.980282322571368, −8.246101883393991, −7.878637270429704, −7.344271997266259, −6.664521451451111, −6.145405960864657, −5.932115589150609, −5.387432487047839, −4.787451983617110, −4.077765294001066, −3.663758459944785, −2.966004395949317, −2.500233238644202, −1.773128125111823, −1.224070499426409, −0.5457358734490538,
0.5457358734490538, 1.224070499426409, 1.773128125111823, 2.500233238644202, 2.966004395949317, 3.663758459944785, 4.077765294001066, 4.787451983617110, 5.387432487047839, 5.932115589150609, 6.145405960864657, 6.664521451451111, 7.344271997266259, 7.878637270429704, 8.246101883393991, 8.980282322571368, 9.302148072086691, 9.712760632118619, 10.28630973765073, 10.72801391946115, 10.94499639741320, 11.80996171762984, 12.16879023318302, 12.61459984818849, 13.11330624362387
