| L(s) = 1 | + 2-s + 4-s + 4·5-s − 2·7-s + 8-s + 4·10-s + 6·13-s − 2·14-s + 16-s + 7·17-s + 4·20-s + 23-s + 11·25-s + 6·26-s − 2·28-s + 5·29-s + 11·31-s + 32-s + 7·34-s − 8·35-s − 5·37-s + 4·40-s − 8·41-s − 43-s + 46-s − 2·47-s − 3·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.78·5-s − 0.755·7-s + 0.353·8-s + 1.26·10-s + 1.66·13-s − 0.534·14-s + 1/4·16-s + 1.69·17-s + 0.894·20-s + 0.208·23-s + 11/5·25-s + 1.17·26-s − 0.377·28-s + 0.928·29-s + 1.97·31-s + 0.176·32-s + 1.20·34-s − 1.35·35-s − 0.821·37-s + 0.632·40-s − 1.24·41-s − 0.152·43-s + 0.147·46-s − 0.291·47-s − 3/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 149454 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149454 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.035328861\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.035328861\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 - 11 T + p T^{2} \) | 1.31.al |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - T + p T^{2} \) | 1.59.ab |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 13 T + p T^{2} \) | 1.73.n |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−13.43241034778613, −13.13468529131861, −12.42545821073484, −12.07916251835199, −11.60510582553813, −10.78756439507442, −10.37801518497516, −10.08644661393687, −9.694047808165544, −9.053301059446149, −8.509012070900626, −8.127639546780507, −7.273794586201165, −6.620266425065131, −6.288885144659896, −6.052407752308420, −5.407035331878299, −5.054209800375199, −4.356739209645521, −3.485943978702186, −3.177938506452082, −2.710771197070920, −1.866530211369098, −1.314689906371026, −0.8419784763471912,
0.8419784763471912, 1.314689906371026, 1.866530211369098, 2.710771197070920, 3.177938506452082, 3.485943978702186, 4.356739209645521, 5.054209800375199, 5.407035331878299, 6.052407752308420, 6.288885144659896, 6.620266425065131, 7.273794586201165, 8.127639546780507, 8.509012070900626, 9.053301059446149, 9.694047808165544, 10.08644661393687, 10.37801518497516, 10.78756439507442, 11.60510582553813, 12.07916251835199, 12.42545821073484, 13.13468529131861, 13.43241034778613
