| L(s) = 1 | − 5-s − 3·7-s + 5·11-s + 2·13-s + 6·17-s − 6·19-s + 23-s − 4·25-s − 2·31-s + 3·35-s − 37-s + 5·41-s + 2·43-s + 9·47-s + 2·49-s − 3·53-s − 5·55-s + 4·59-s − 2·61-s − 2·65-s + 2·67-s + 6·71-s − 7·73-s − 15·77-s − 12·79-s − 7·83-s − 6·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.13·7-s + 1.50·11-s + 0.554·13-s + 1.45·17-s − 1.37·19-s + 0.208·23-s − 4/5·25-s − 0.359·31-s + 0.507·35-s − 0.164·37-s + 0.780·41-s + 0.304·43-s + 1.31·47-s + 2/7·49-s − 0.412·53-s − 0.674·55-s + 0.520·59-s − 0.256·61-s − 0.248·65-s + 0.244·67-s + 0.712·71-s − 0.819·73-s − 1.70·77-s − 1.35·79-s − 0.768·83-s − 0.650·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.679673910\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.679673910\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 397 | \( 1 + T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + 7 T + p T^{2} \) | 1.83.h |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−12.81940154944381, −12.36950556229602, −12.21865059999998, −11.44668207262195, −11.27260752379106, −10.48081019439809, −10.22020247980249, −9.520189257644079, −9.284221275322028, −8.785455550602123, −8.228376574767201, −7.769826840018221, −7.091001236138572, −6.787530713144586, −6.180330532617104, −5.875118675114626, −5.372397810685436, −4.386520668905972, −4.008406059758323, −3.719021714438546, −3.114691894554527, −2.535923646123728, −1.673631834721847, −1.137493110622491, −0.3905266280761170,
0.3905266280761170, 1.137493110622491, 1.673631834721847, 2.535923646123728, 3.114691894554527, 3.719021714438546, 4.008406059758323, 4.386520668905972, 5.372397810685436, 5.875118675114626, 6.180330532617104, 6.787530713144586, 7.091001236138572, 7.769826840018221, 8.228376574767201, 8.785455550602123, 9.284221275322028, 9.520189257644079, 10.22020247980249, 10.48081019439809, 11.27260752379106, 11.44668207262195, 12.21865059999998, 12.36950556229602, 12.81940154944381
