| L(s) = 1 | + 2·2-s + 2·4-s + 3·7-s − 13-s + 6·14-s − 4·16-s + 2·17-s + 5·19-s − 6·23-s − 2·26-s + 6·28-s + 10·29-s − 3·31-s − 8·32-s + 4·34-s + 2·37-s + 10·38-s − 8·41-s − 43-s − 12·46-s − 2·47-s + 2·49-s − 2·52-s + 4·53-s + 20·58-s + 10·59-s − 7·61-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 1.13·7-s − 0.277·13-s + 1.60·14-s − 16-s + 0.485·17-s + 1.14·19-s − 1.25·23-s − 0.392·26-s + 1.13·28-s + 1.85·29-s − 0.538·31-s − 1.41·32-s + 0.685·34-s + 0.328·37-s + 1.62·38-s − 1.24·41-s − 0.152·43-s − 1.76·46-s − 0.291·47-s + 2/7·49-s − 0.277·52-s + 0.549·53-s + 2.62·58-s + 1.30·59-s − 0.896·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.859620764\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.859620764\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 2 | \( 1 - p T + p T^{2} \) | 1.2.ac |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 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 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 17 T + p T^{2} \) | 1.97.ar |
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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
−15.10597467852376, −14.54224401881932, −14.08766790804382, −13.87880584553911, −13.26757711893551, −12.54259504844652, −12.03280578741956, −11.75367098075227, −11.27682592021489, −10.48066788959873, −9.964912409374768, −9.308352170085874, −8.477186363713402, −8.068771979813621, −7.404354739687299, −6.747414311960389, −6.091306411451846, −5.489049223321053, −4.896427432808544, −4.642283634780656, −3.736439733866454, −3.284854025964504, −2.417021690744215, −1.781391951651428, −0.7525081589095767,
0.7525081589095767, 1.781391951651428, 2.417021690744215, 3.284854025964504, 3.736439733866454, 4.642283634780656, 4.896427432808544, 5.489049223321053, 6.091306411451846, 6.747414311960389, 7.404354739687299, 8.068771979813621, 8.477186363713402, 9.308352170085874, 9.964912409374768, 10.48066788959873, 11.27682592021489, 11.75367098075227, 12.03280578741956, 12.54259504844652, 13.26757711893551, 13.87880584553911, 14.08766790804382, 14.54224401881932, 15.10597467852376
