| L(s) = 1 | − 2-s + 4-s − 5-s + 7-s − 8-s + 10-s + 6·11-s + 2·13-s − 14-s + 16-s − 4·19-s − 20-s − 6·22-s + 9·23-s + 25-s − 2·26-s + 28-s + 3·29-s + 4·31-s − 32-s − 35-s − 8·37-s + 4·38-s + 40-s − 3·41-s + 8·43-s + 6·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s − 0.353·8-s + 0.316·10-s + 1.80·11-s + 0.554·13-s − 0.267·14-s + 1/4·16-s − 0.917·19-s − 0.223·20-s − 1.27·22-s + 1.87·23-s + 1/5·25-s − 0.392·26-s + 0.188·28-s + 0.557·29-s + 0.718·31-s − 0.176·32-s − 0.169·35-s − 1.31·37-s + 0.648·38-s + 0.158·40-s − 0.468·41-s + 1.21·43-s + 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 234090 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234090 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.382993502\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.382993502\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 13 T + p T^{2} \) | 1.61.an |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 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
−12.75425904524127, −12.34416106339683, −11.85300013025791, −11.49425211093107, −11.03751943183979, −10.62809668572193, −10.24302878653041, −9.338957817438657, −9.208493633017596, −8.735150405151255, −8.299275061199362, −7.880163348583665, −7.046022077240940, −6.884668692835742, −6.403861671543169, −5.902800697054872, −5.153036799668064, −4.537223499771256, −4.182562506861390, −3.430921416691547, −3.120353633610547, −2.269407891170373, −1.563495699955090, −1.145436340579170, −0.5383525008582212,
0.5383525008582212, 1.145436340579170, 1.563495699955090, 2.269407891170373, 3.120353633610547, 3.430921416691547, 4.182562506861390, 4.537223499771256, 5.153036799668064, 5.902800697054872, 6.403861671543169, 6.884668692835742, 7.046022077240940, 7.880163348583665, 8.299275061199362, 8.735150405151255, 9.208493633017596, 9.338957817438657, 10.24302878653041, 10.62809668572193, 11.03751943183979, 11.49425211093107, 11.85300013025791, 12.34416106339683, 12.75425904524127
