| L(s) = 1 | − 2·3-s − 7-s + 9-s − 6·13-s + 6·17-s + 4·19-s + 2·21-s + 4·27-s − 4·29-s + 2·31-s + 6·37-s + 12·39-s + 2·41-s − 4·43-s + 10·47-s + 49-s − 12·51-s + 6·53-s − 8·57-s − 6·59-s − 14·61-s − 63-s − 12·67-s + 8·71-s + 10·73-s − 12·79-s − 11·81-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.377·7-s + 1/3·9-s − 1.66·13-s + 1.45·17-s + 0.917·19-s + 0.436·21-s + 0.769·27-s − 0.742·29-s + 0.359·31-s + 0.986·37-s + 1.92·39-s + 0.312·41-s − 0.609·43-s + 1.45·47-s + 1/7·49-s − 1.68·51-s + 0.824·53-s − 1.05·57-s − 0.781·59-s − 1.79·61-s − 0.125·63-s − 1.46·67-s + 0.949·71-s + 1.17·73-s − 1.35·79-s − 1.22·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.110143363\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.110143363\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 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
−12.41646007092435, −12.03051911869218, −11.84481222830548, −11.32457463859285, −10.71429101067060, −10.33679932953817, −9.948035642363723, −9.427107705135812, −9.194078005932537, −8.401046897944621, −7.692309275423574, −7.489228369970641, −7.116014309890423, −6.347990129319844, −6.019342205149238, −5.488897469514479, −5.181791726281998, −4.678054714414363, −4.136925940993377, −3.390916256507443, −2.900152650188250, −2.439516554087100, −1.570630588753503, −0.8962314698354741, −0.3729434767070306,
0.3729434767070306, 0.8962314698354741, 1.570630588753503, 2.439516554087100, 2.900152650188250, 3.390916256507443, 4.136925940993377, 4.678054714414363, 5.181791726281998, 5.488897469514479, 6.019342205149238, 6.347990129319844, 7.116014309890423, 7.489228369970641, 7.692309275423574, 8.401046897944621, 9.194078005932537, 9.427107705135812, 9.948035642363723, 10.33679932953817, 10.71429101067060, 11.32457463859285, 11.84481222830548, 12.03051911869218, 12.41646007092435
