| L(s) = 1 | − 3-s − 5-s + 2·7-s − 2·9-s + 11-s + 15-s − 6·19-s − 2·21-s + 5·23-s − 4·25-s + 5·27-s + 2·29-s + 5·31-s − 33-s − 2·35-s − 3·37-s + 2·41-s + 4·43-s + 2·45-s + 12·47-s − 3·49-s + 6·53-s − 55-s + 6·57-s − 3·59-s − 14·61-s − 4·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.755·7-s − 2/3·9-s + 0.301·11-s + 0.258·15-s − 1.37·19-s − 0.436·21-s + 1.04·23-s − 4/5·25-s + 0.962·27-s + 0.371·29-s + 0.898·31-s − 0.174·33-s − 0.338·35-s − 0.493·37-s + 0.312·41-s + 0.609·43-s + 0.298·45-s + 1.75·47-s − 3/7·49-s + 0.824·53-s − 0.134·55-s + 0.794·57-s − 0.390·59-s − 1.79·61-s − 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 118976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 118976 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.134160437\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.134160437\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 + 12 T + p T^{2} \) | 1.73.m |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 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
−13.53985468645466, −13.12862270705155, −12.37696085633465, −12.01770720919525, −11.75929126098730, −11.05537229158189, −10.81146597902167, −10.43116295951271, −9.689877519451504, −8.981301251146363, −8.645725356738388, −8.253034554634060, −7.590776317937022, −7.139296663460460, −6.529670413085330, −5.915043381124798, −5.652922308886801, −4.859138866875588, −4.370703659160488, −4.082224946930114, −3.065543507517387, −2.688785658125951, −1.844321460891652, −1.162536517549892, −0.3642637715854704,
0.3642637715854704, 1.162536517549892, 1.844321460891652, 2.688785658125951, 3.065543507517387, 4.082224946930114, 4.370703659160488, 4.859138866875588, 5.652922308886801, 5.915043381124798, 6.529670413085330, 7.139296663460460, 7.590776317937022, 8.253034554634060, 8.645725356738388, 8.981301251146363, 9.689877519451504, 10.43116295951271, 10.81146597902167, 11.05537229158189, 11.75929126098730, 12.01770720919525, 12.37696085633465, 13.12862270705155, 13.53985468645466
