| L(s) = 1 | − 2·3-s + 9-s − 11-s − 2·13-s + 4·17-s + 5·23-s + 4·27-s − 3·29-s − 10·31-s + 2·33-s + 5·37-s + 4·39-s − 10·41-s − 5·43-s − 4·47-s − 8·51-s + 10·53-s − 10·59-s + 10·61-s + 5·67-s − 10·69-s + 3·71-s − 10·73-s + 13·79-s − 11·81-s − 10·83-s + 6·87-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/3·9-s − 0.301·11-s − 0.554·13-s + 0.970·17-s + 1.04·23-s + 0.769·27-s − 0.557·29-s − 1.79·31-s + 0.348·33-s + 0.821·37-s + 0.640·39-s − 1.56·41-s − 0.762·43-s − 0.583·47-s − 1.12·51-s + 1.37·53-s − 1.30·59-s + 1.28·61-s + 0.610·67-s − 1.20·69-s + 0.356·71-s − 1.17·73-s + 1.46·79-s − 1.22·81-s − 1.09·83-s + 0.643·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8756798750\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8756798750\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 13 T + p T^{2} \) | 1.79.an |
| 83 | \( 1 + 10 T + p T^{2} \) | 1.83.k |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−8.184282943321720352850557978661, −7.36139730829428658231446957892, −6.82681189579378952219983828801, −5.95756921319978391985513022225, −5.27152300145816091696474125824, −4.95210295579370175451769344034, −3.76943144689912104706477866261, −2.93143676363544727575723759262, −1.73292135759538157393815956866, −0.54393249968075621364458993818,
0.54393249968075621364458993818, 1.73292135759538157393815956866, 2.93143676363544727575723759262, 3.76943144689912104706477866261, 4.95210295579370175451769344034, 5.27152300145816091696474125824, 5.95756921319978391985513022225, 6.82681189579378952219983828801, 7.36139730829428658231446957892, 8.184282943321720352850557978661
