| L(s) = 1 | − 3·3-s + 3·5-s − 2·7-s + 6·9-s − 11-s − 2·13-s − 9·15-s − 19-s + 6·21-s − 23-s + 4·25-s − 9·27-s + 3·29-s − 8·31-s + 3·33-s − 6·35-s − 8·37-s + 6·39-s − 6·41-s − 7·43-s + 18·45-s − 7·47-s − 3·49-s + 12·53-s − 3·55-s + 3·57-s − 11·59-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1.34·5-s − 0.755·7-s + 2·9-s − 0.301·11-s − 0.554·13-s − 2.32·15-s − 0.229·19-s + 1.30·21-s − 0.208·23-s + 4/5·25-s − 1.73·27-s + 0.557·29-s − 1.43·31-s + 0.522·33-s − 1.01·35-s − 1.31·37-s + 0.960·39-s − 0.937·41-s − 1.06·43-s + 2.68·45-s − 1.02·47-s − 3/7·49-s + 1.64·53-s − 0.404·55-s + 0.397·57-s − 1.43·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4832205852\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4832205852\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| 23 | \( 1 + T \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 11 T + p T^{2} \) | 1.59.l |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 5 T + p T^{2} \) | 1.73.f |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 - 11 T + p T^{2} \) | 1.89.al |
| 97 | \( 1 + 9 T + p T^{2} \) | 1.97.j |
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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
−15.24742894733177, −14.83067071781552, −13.98820860133047, −13.40747832800555, −13.10541955759987, −12.44655333990757, −12.10526297598154, −11.50994098402586, −10.76263766092483, −10.36390559341201, −10.01192204207374, −9.484871875842677, −8.887203509456947, −7.981806732656156, −7.086744666022899, −6.687390560702822, −6.285463913689970, −5.585854183805540, −5.264336999242147, −4.756024824496463, −3.813502895459258, −2.987298641837658, −1.994850441414444, −1.477692111861786, −0.2961412038889810,
0.2961412038889810, 1.477692111861786, 1.994850441414444, 2.987298641837658, 3.813502895459258, 4.756024824496463, 5.264336999242147, 5.585854183805540, 6.285463913689970, 6.687390560702822, 7.086744666022899, 7.981806732656156, 8.887203509456947, 9.484871875842677, 10.01192204207374, 10.36390559341201, 10.76263766092483, 11.50994098402586, 12.10526297598154, 12.44655333990757, 13.10541955759987, 13.40747832800555, 13.98820860133047, 14.83067071781552, 15.24742894733177
