| L(s) = 1 | + 2-s + 4-s − 2·5-s + 8-s − 2·10-s − 2·11-s − 4·13-s + 16-s + 19-s − 2·20-s − 2·22-s − 25-s − 4·26-s − 2·29-s − 2·31-s + 32-s + 8·37-s + 38-s − 2·40-s − 2·41-s − 4·43-s − 2·44-s − 4·47-s − 7·49-s − 50-s − 4·52-s − 2·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.353·8-s − 0.632·10-s − 0.603·11-s − 1.10·13-s + 1/4·16-s + 0.229·19-s − 0.447·20-s − 0.426·22-s − 1/5·25-s − 0.784·26-s − 0.371·29-s − 0.359·31-s + 0.176·32-s + 1.31·37-s + 0.162·38-s − 0.316·40-s − 0.312·41-s − 0.609·43-s − 0.301·44-s − 0.583·47-s − 49-s − 0.141·50-s − 0.554·52-s − 0.274·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180918 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180918 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9581144918\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9581144918\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 19 | \( 1 - T \) | |
| 23 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−13.19015868878840, −12.77186918555781, −12.14984470744976, −11.79509614092128, −11.43179929501610, −10.99545052969360, −10.32297379625581, −9.964760522660315, −9.430948704572275, −8.850307728759136, −8.091807643195402, −7.781457573350448, −7.501809288704839, −6.781901586103622, −6.462645459862876, −5.564218068544390, −5.364995293155808, −4.593402460561960, −4.390821265324514, −3.621040736542870, −3.201353566683510, −2.578624253549772, −2.044999358061872, −1.243559094928580, −0.2457173507575646,
0.2457173507575646, 1.243559094928580, 2.044999358061872, 2.578624253549772, 3.201353566683510, 3.621040736542870, 4.390821265324514, 4.593402460561960, 5.364995293155808, 5.564218068544390, 6.462645459862876, 6.781901586103622, 7.501809288704839, 7.781457573350448, 8.091807643195402, 8.850307728759136, 9.430948704572275, 9.964760522660315, 10.32297379625581, 10.99545052969360, 11.43179929501610, 11.79509614092128, 12.14984470744976, 12.77186918555781, 13.19015868878840
