| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 8-s + 9-s − 10-s + 2·11-s + 12-s + 15-s + 16-s − 18-s + 20-s − 2·22-s − 24-s + 25-s + 27-s − 30-s + 2·31-s − 32-s + 2·33-s + 36-s + 8·37-s − 40-s + 10·41-s + 2·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.603·11-s + 0.288·12-s + 0.258·15-s + 1/4·16-s − 0.235·18-s + 0.223·20-s − 0.426·22-s − 0.204·24-s + 1/5·25-s + 0.192·27-s − 0.182·30-s + 0.359·31-s − 0.176·32-s + 0.348·33-s + 1/6·36-s + 1.31·37-s − 0.158·40-s + 1.56·41-s + 0.304·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.725153926\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.725153926\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 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 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−12.81122987904953, −12.52532837776235, −11.80202928069894, −11.35489558032349, −11.07441326670215, −10.32376965819419, −9.997127009419059, −9.591329758365577, −9.028077578023962, −8.874313372926245, −8.130869723143694, −7.826096805347162, −7.293305075036236, −6.739494920981345, −6.300013514021146, −5.847743599767504, −5.235480218001119, −4.508968645587107, −4.101922856839794, −3.396433348514739, −2.897968566855505, −2.233309129088555, −1.899908607989532, −1.014824706777865, −0.6652898962858526,
0.6652898962858526, 1.014824706777865, 1.899908607989532, 2.233309129088555, 2.897968566855505, 3.396433348514739, 4.101922856839794, 4.508968645587107, 5.235480218001119, 5.847743599767504, 6.300013514021146, 6.739494920981345, 7.293305075036236, 7.826096805347162, 8.130869723143694, 8.874313372926245, 9.028077578023962, 9.591329758365577, 9.997127009419059, 10.32376965819419, 11.07441326670215, 11.35489558032349, 11.80202928069894, 12.52532837776235, 12.81122987904953
