| L(s) = 1 | − 3-s + 5-s + 9-s + 2·11-s + 4·13-s − 15-s + 25-s − 27-s + 2·29-s − 10·31-s − 2·33-s + 4·37-s − 4·39-s + 6·41-s + 4·43-s + 45-s − 7·49-s + 2·53-s + 2·55-s + 6·59-s + 6·61-s + 4·65-s + 4·67-s − 8·71-s + 14·73-s − 75-s − 2·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.603·11-s + 1.10·13-s − 0.258·15-s + 1/5·25-s − 0.192·27-s + 0.371·29-s − 1.79·31-s − 0.348·33-s + 0.657·37-s − 0.640·39-s + 0.937·41-s + 0.609·43-s + 0.149·45-s − 49-s + 0.274·53-s + 0.269·55-s + 0.781·59-s + 0.768·61-s + 0.496·65-s + 0.488·67-s − 0.949·71-s + 1.63·73-s − 0.115·75-s − 0.225·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.706340990\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.706340990\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−9.253995078851749104832706258794, −8.532597792034447590591958578745, −7.54733984797030655324096164336, −6.66734796873224642317270126902, −6.02414119331468170949153182777, −5.33349421728964499565168926647, −4.26623619018282517433376087877, −3.44960428840436628253207297190, −2.06002223875355254437768868122, −0.951853117585691762831532111621,
0.951853117585691762831532111621, 2.06002223875355254437768868122, 3.44960428840436628253207297190, 4.26623619018282517433376087877, 5.33349421728964499565168926647, 6.02414119331468170949153182777, 6.66734796873224642317270126902, 7.54733984797030655324096164336, 8.532597792034447590591958578745, 9.253995078851749104832706258794
