| L(s) = 1 | + 3·5-s − 4·7-s − 3·11-s − 4·13-s − 6·17-s − 19-s + 23-s + 4·25-s + 2·31-s − 12·35-s − 37-s + 2·41-s + 4·43-s + 10·47-s + 9·49-s + 3·53-s − 9·55-s + 3·59-s − 5·61-s − 12·65-s + 2·67-s + 9·71-s − 11·73-s + 12·77-s − 10·79-s − 3·83-s − 18·85-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 1.51·7-s − 0.904·11-s − 1.10·13-s − 1.45·17-s − 0.229·19-s + 0.208·23-s + 4/5·25-s + 0.359·31-s − 2.02·35-s − 0.164·37-s + 0.312·41-s + 0.609·43-s + 1.45·47-s + 9/7·49-s + 0.412·53-s − 1.21·55-s + 0.390·59-s − 0.640·61-s − 1.48·65-s + 0.244·67-s + 1.06·71-s − 1.28·73-s + 1.36·77-s − 1.12·79-s − 0.329·83-s − 1.95·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 101232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6149798384\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6149798384\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| 37 | \( 1 + T \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 + 13 T + p T^{2} \) | 1.89.n |
| 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
−13.56073291296584, −13.27256120100721, −12.91460773222058, −12.40766622129622, −12.01373646519445, −11.07875147694492, −10.70768955105287, −10.14580638567050, −9.845006482382150, −9.366412229648530, −8.967672104973862, −8.425373956082144, −7.560660719594615, −7.099097056176975, −6.592300721652672, −6.192983810319764, −5.554834409105893, −5.241633502877874, −4.418787448233782, −3.944105247359411, −2.920192301614630, −2.543263687321980, −2.297521617301976, −1.304514666410534, −0.2355120743020208,
0.2355120743020208, 1.304514666410534, 2.297521617301976, 2.543263687321980, 2.920192301614630, 3.944105247359411, 4.418787448233782, 5.241633502877874, 5.554834409105893, 6.192983810319764, 6.592300721652672, 7.099097056176975, 7.560660719594615, 8.425373956082144, 8.967672104973862, 9.366412229648530, 9.845006482382150, 10.14580638567050, 10.70768955105287, 11.07875147694492, 12.01373646519445, 12.40766622129622, 12.91460773222058, 13.27256120100721, 13.56073291296584
