| L(s) = 1 | + 11-s − 2·13-s − 6·17-s − 2·19-s + 4·23-s + 2·29-s + 4·31-s + 8·37-s − 2·41-s − 12·43-s + 8·47-s − 7·49-s − 6·53-s + 14·61-s − 12·67-s + 12·71-s + 6·73-s − 6·79-s − 14·83-s + 12·97-s + 14·101-s + 4·103-s − 6·107-s − 10·109-s + 18·113-s + ⋯ |
| L(s) = 1 | + 0.301·11-s − 0.554·13-s − 1.45·17-s − 0.458·19-s + 0.834·23-s + 0.371·29-s + 0.718·31-s + 1.31·37-s − 0.312·41-s − 1.82·43-s + 1.16·47-s − 49-s − 0.824·53-s + 1.79·61-s − 1.46·67-s + 1.42·71-s + 0.702·73-s − 0.675·79-s − 1.53·83-s + 1.21·97-s + 1.39·101-s + 0.394·103-s − 0.580·107-s − 0.957·109-s + 1.69·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.670693258\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.670693258\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−7.65743840324001735248618925241, −6.75804173648619847761297178475, −6.56734540209241168953766149104, −5.62331116299908881721176740356, −4.71114192225086257340647695389, −4.40678518450235365260112721170, −3.35908569236542002649809298669, −2.57234558949746491754476582116, −1.79628078042050405972321124715, −0.60204989245330181610578372085,
0.60204989245330181610578372085, 1.79628078042050405972321124715, 2.57234558949746491754476582116, 3.35908569236542002649809298669, 4.40678518450235365260112721170, 4.71114192225086257340647695389, 5.62331116299908881721176740356, 6.56734540209241168953766149104, 6.75804173648619847761297178475, 7.65743840324001735248618925241
