| L(s) = 1 | + 2·3-s + 9-s − 11-s + 13-s − 2·17-s − 8·19-s − 6·23-s − 4·27-s + 2·29-s + 7·31-s − 2·33-s + 8·37-s + 2·39-s − 8·41-s − 3·43-s − 12·47-s − 4·51-s + 10·53-s − 16·57-s + 3·59-s − 6·61-s + 4·67-s − 12·69-s − 71-s − 13·73-s − 4·79-s − 11·81-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s − 0.301·11-s + 0.277·13-s − 0.485·17-s − 1.83·19-s − 1.25·23-s − 0.769·27-s + 0.371·29-s + 1.25·31-s − 0.348·33-s + 1.31·37-s + 0.320·39-s − 1.24·41-s − 0.457·43-s − 1.75·47-s − 0.560·51-s + 1.37·53-s − 2.11·57-s + 0.390·59-s − 0.768·61-s + 0.488·67-s − 1.44·69-s − 0.118·71-s − 1.52·73-s − 0.450·79-s − 1.22·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 215600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 215600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.535317773\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.535317773\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + 3 T + p T^{2} \) | 1.43.d |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + T + p T^{2} \) | 1.71.b |
| 73 | \( 1 + 13 T + p T^{2} \) | 1.73.n |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 + T + p T^{2} \) | 1.89.b |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−13.26626115496501, −12.68372812400501, −11.97991959989535, −11.71115383585027, −11.08960255238418, −10.57391363384742, −10.02109131278628, −9.824319310657011, −9.091793732650616, −8.578326717576856, −8.320148352358227, −8.034812612904029, −7.416738083848257, −6.676586910540551, −6.371553157635613, −5.877316811041724, −5.157269172263110, −4.407231955068986, −4.220767048447707, −3.542249780209813, −2.948263439530645, −2.434841661959864, −2.009812996842815, −1.378713512858220, −0.2972557225891795,
0.2972557225891795, 1.378713512858220, 2.009812996842815, 2.434841661959864, 2.948263439530645, 3.542249780209813, 4.220767048447707, 4.407231955068986, 5.157269172263110, 5.877316811041724, 6.371553157635613, 6.676586910540551, 7.416738083848257, 8.034812612904029, 8.320148352358227, 8.578326717576856, 9.091793732650616, 9.824319310657011, 10.02109131278628, 10.57391363384742, 11.08960255238418, 11.71115383585027, 11.97991959989535, 12.68372812400501, 13.26626115496501
