| L(s) = 1 | − 2·3-s + 5-s + 9-s − 11-s + 13-s − 2·15-s − 2·17-s + 6·23-s + 25-s + 4·27-s − 2·29-s + 3·31-s + 2·33-s + 8·37-s − 2·39-s + 43-s + 45-s − 8·47-s + 4·51-s − 6·53-s − 55-s + 3·59-s − 2·61-s + 65-s − 8·67-s − 12·69-s + 3·71-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s + 1/3·9-s − 0.301·11-s + 0.277·13-s − 0.516·15-s − 0.485·17-s + 1.25·23-s + 1/5·25-s + 0.769·27-s − 0.371·29-s + 0.538·31-s + 0.348·33-s + 1.31·37-s − 0.320·39-s + 0.152·43-s + 0.149·45-s − 1.16·47-s + 0.560·51-s − 0.824·53-s − 0.134·55-s + 0.390·59-s − 0.256·61-s + 0.124·65-s − 0.977·67-s − 1.44·69-s + 0.356·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.215252229\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.215252229\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 - 7 T + p T^{2} \) | 1.89.ah |
| 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
−16.63544587704824, −16.13080352561183, −15.47157916703177, −14.87400627177570, −14.26767931991792, −13.53252300503095, −12.94879365078881, −12.63610976954072, −11.66472977117808, −11.37922152663846, −10.80071775636401, −10.26501502955247, −9.562521265209285, −8.925664926495908, −8.258964244611212, −7.430336070246587, −6.736200909353114, −6.153035717480830, −5.682824919640352, −4.880256728541810, −4.497149583517459, −3.308305490362354, −2.578276971418592, −1.495538756749576, −0.5692062302265030,
0.5692062302265030, 1.495538756749576, 2.578276971418592, 3.308305490362354, 4.497149583517459, 4.880256728541810, 5.682824919640352, 6.153035717480830, 6.736200909353114, 7.430336070246587, 8.258964244611212, 8.925664926495908, 9.562521265209285, 10.26501502955247, 10.80071775636401, 11.37922152663846, 11.66472977117808, 12.63610976954072, 12.94879365078881, 13.53252300503095, 14.26767931991792, 14.87400627177570, 15.47157916703177, 16.13080352561183, 16.63544587704824
