| L(s) = 1 | − 3-s − 4·5-s − 2·9-s + 3·11-s + 5·13-s + 4·15-s − 5·19-s − 6·23-s + 11·25-s + 5·27-s + 5·29-s − 6·31-s − 3·33-s − 6·37-s − 5·39-s − 12·41-s − 3·43-s + 8·45-s − 2·47-s − 7·49-s − 2·53-s − 12·55-s + 5·57-s − 4·59-s − 13·61-s − 20·65-s + 12·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.78·5-s − 2/3·9-s + 0.904·11-s + 1.38·13-s + 1.03·15-s − 1.14·19-s − 1.25·23-s + 11/5·25-s + 0.962·27-s + 0.928·29-s − 1.07·31-s − 0.522·33-s − 0.986·37-s − 0.800·39-s − 1.87·41-s − 0.457·43-s + 1.19·45-s − 0.291·47-s − 49-s − 0.274·53-s − 1.61·55-s + 0.662·57-s − 0.520·59-s − 1.66·61-s − 2.48·65-s + 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 383792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 383792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 17 | \( 1 \) | |
| 83 | \( 1 - T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + 3 T + p T^{2} \) | 1.43.d |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 + 5 T + p T^{2} \) | 1.73.f |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−12.48504803660151, −12.04447891655043, −11.80894488066252, −11.42202000279249, −11.00666685752993, −10.58957648694536, −10.27787980125873, −9.406236410156448, −8.878535539645775, −8.452230539266781, −8.258875759156075, −7.843400604204278, −7.041584399219808, −6.670649884208268, −6.323968794841251, −5.859911002435337, −5.111867061782986, −4.719054880005502, −4.032600787568707, −3.831051698666676, −3.303809664143655, −2.847181453968776, −1.738161987091542, −1.419782212381394, −0.4427781542713456, 0,
0.4427781542713456, 1.419782212381394, 1.738161987091542, 2.847181453968776, 3.303809664143655, 3.831051698666676, 4.032600787568707, 4.719054880005502, 5.111867061782986, 5.859911002435337, 6.323968794841251, 6.670649884208268, 7.041584399219808, 7.843400604204278, 8.258875759156075, 8.452230539266781, 8.878535539645775, 9.406236410156448, 10.27787980125873, 10.58957648694536, 11.00666685752993, 11.42202000279249, 11.80894488066252, 12.04447891655043, 12.48504803660151
