| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s − 3·7-s + 8-s + 9-s − 10-s + 6·11-s − 12-s − 3·14-s + 15-s + 16-s + 4·17-s + 18-s − 19-s − 20-s + 3·21-s + 6·22-s − 23-s − 24-s + 25-s − 27-s − 3·28-s + 30-s + 3·31-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 1.13·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.80·11-s − 0.288·12-s − 0.801·14-s + 0.258·15-s + 1/4·16-s + 0.970·17-s + 0.235·18-s − 0.229·19-s − 0.223·20-s + 0.654·21-s + 1.27·22-s − 0.208·23-s − 0.204·24-s + 1/5·25-s − 0.192·27-s − 0.566·28-s + 0.182·30-s + 0.538·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.182891292\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.182891292\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 5 T + p T^{2} \) | 1.47.af |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 - 13 T + p T^{2} \) | 1.59.an |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 97 | \( 1 + 15 T + p T^{2} \) | 1.97.p |
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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
−13.85953806995720, −13.24230666325878, −12.65560752888130, −12.23879652273719, −12.06647888538770, −11.44458264470363, −11.06610677264028, −10.34417965761810, −9.852585130594953, −9.545410559565274, −8.744154866995794, −8.392111822473319, −7.435048755794214, −7.098096412443080, −6.529715599734428, −6.252577088050380, −5.623777041618059, −5.104777420100257, −4.324084096425796, −3.773083214906407, −3.606455886613850, −2.829309495140900, −2.007963720758391, −1.160627729499868, −0.5818334745039002,
0.5818334745039002, 1.160627729499868, 2.007963720758391, 2.829309495140900, 3.606455886613850, 3.773083214906407, 4.324084096425796, 5.104777420100257, 5.623777041618059, 6.252577088050380, 6.529715599734428, 7.098096412443080, 7.435048755794214, 8.392111822473319, 8.744154866995794, 9.545410559565274, 9.852585130594953, 10.34417965761810, 11.06610677264028, 11.44458264470363, 12.06647888538770, 12.23879652273719, 12.65560752888130, 13.24230666325878, 13.85953806995720
