| L(s) = 1 | − 2·3-s + 5-s + 9-s − 2·11-s − 2·15-s + 2·17-s + 4·19-s − 4·23-s + 25-s + 4·27-s + 4·29-s − 31-s + 4·33-s + 8·37-s + 6·41-s − 2·43-s + 45-s − 7·49-s − 4·51-s − 8·53-s − 2·55-s − 8·57-s − 8·59-s − 4·67-s + 8·69-s + 6·73-s − 2·75-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s + 1/3·9-s − 0.603·11-s − 0.516·15-s + 0.485·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s + 0.769·27-s + 0.742·29-s − 0.179·31-s + 0.696·33-s + 1.31·37-s + 0.937·41-s − 0.304·43-s + 0.149·45-s − 49-s − 0.560·51-s − 1.09·53-s − 0.269·55-s − 1.05·57-s − 1.04·59-s − 0.488·67-s + 0.963·69-s + 0.702·73-s − 0.230·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.181607303\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.181607303\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 31 | \( 1 + T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−7.64922737077090565350897186292, −6.84037272626182182026356872752, −5.94939111103903493515414346843, −5.89408266018881514174170183139, −4.95638569372654383517660061859, −4.52030469160325774192844738108, −3.33472876047482802668772688457, −2.62625624888861871112114572958, −1.51479246882211507099739763644, −0.56810740418014069389079114466,
0.56810740418014069389079114466, 1.51479246882211507099739763644, 2.62625624888861871112114572958, 3.33472876047482802668772688457, 4.52030469160325774192844738108, 4.95638569372654383517660061859, 5.89408266018881514174170183139, 5.94939111103903493515414346843, 6.84037272626182182026356872752, 7.64922737077090565350897186292
