| L(s) = 1 | − 5-s − 2·7-s − 2·11-s + 6·17-s − 19-s − 8·23-s + 25-s − 6·29-s + 4·31-s + 2·35-s + 4·37-s − 4·41-s − 12·43-s − 3·49-s + 14·53-s + 2·55-s + 10·59-s + 10·61-s + 4·67-s − 4·71-s − 2·73-s + 4·77-s + 4·79-s − 12·83-s − 6·85-s + 95-s + 6·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s − 0.603·11-s + 1.45·17-s − 0.229·19-s − 1.66·23-s + 1/5·25-s − 1.11·29-s + 0.718·31-s + 0.338·35-s + 0.657·37-s − 0.624·41-s − 1.82·43-s − 3/7·49-s + 1.92·53-s + 0.269·55-s + 1.30·59-s + 1.28·61-s + 0.488·67-s − 0.474·71-s − 0.234·73-s + 0.455·77-s + 0.450·79-s − 1.31·83-s − 0.650·85-s + 0.102·95-s + 0.609·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.130782853\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.130782853\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 19 | \( 1 + T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 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
−8.162745214792923739155905108121, −7.25582622219193369343544421719, −6.64682065446711976337156247606, −5.74986303951624470288841879480, −5.31524106705594833928278262866, −4.19455381555038883270515877012, −3.59991679030690289590493265772, −2.85416740088716720595893922391, −1.85300750014569949457401706644, −0.52811467918742385746033645858,
0.52811467918742385746033645858, 1.85300750014569949457401706644, 2.85416740088716720595893922391, 3.59991679030690289590493265772, 4.19455381555038883270515877012, 5.31524106705594833928278262866, 5.74986303951624470288841879480, 6.64682065446711976337156247606, 7.25582622219193369343544421719, 8.162745214792923739155905108121
