| L(s) = 1 | + 3-s + 9-s − 4·11-s + 6·13-s − 2·17-s + 8·19-s + 4·23-s + 27-s − 6·29-s + 4·31-s − 4·33-s − 2·37-s + 6·39-s + 2·41-s + 12·43-s − 2·51-s + 2·53-s + 8·57-s + 4·59-s + 6·61-s + 4·67-s + 4·69-s − 8·71-s + 6·73-s + 16·79-s + 81-s − 4·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 1.20·11-s + 1.66·13-s − 0.485·17-s + 1.83·19-s + 0.834·23-s + 0.192·27-s − 1.11·29-s + 0.718·31-s − 0.696·33-s − 0.328·37-s + 0.960·39-s + 0.312·41-s + 1.82·43-s − 0.280·51-s + 0.274·53-s + 1.05·57-s + 0.520·59-s + 0.768·61-s + 0.488·67-s + 0.481·69-s − 0.949·71-s + 0.702·73-s + 1.80·79-s + 1/9·81-s − 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.726109775\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.726109775\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 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 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 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
−12.94422191177242, −12.74606658654657, −11.82104924705917, −11.56177753440755, −10.98927042157404, −10.57996724275112, −10.22840702621955, −9.488395304959925, −9.081357106742367, −8.854850986125227, −8.071368890696600, −7.789714808634526, −7.394521027400028, −6.759114469450803, −6.240716390844025, −5.563033991330999, −5.309543303032830, −4.672218403194256, −3.962782518699730, −3.518134511066212, −3.051937043884990, −2.462207906104156, −1.895631833970042, −1.027203738552965, −0.6788200022517818,
0.6788200022517818, 1.027203738552965, 1.895631833970042, 2.462207906104156, 3.051937043884990, 3.518134511066212, 3.962782518699730, 4.672218403194256, 5.309543303032830, 5.563033991330999, 6.240716390844025, 6.759114469450803, 7.394521027400028, 7.789714808634526, 8.071368890696600, 8.854850986125227, 9.081357106742367, 9.488395304959925, 10.22840702621955, 10.57996724275112, 10.98927042157404, 11.56177753440755, 11.82104924705917, 12.74606658654657, 12.94422191177242
