| L(s) = 1 | − 3-s − 2·5-s + 7-s − 2·9-s − 5·11-s + 2·15-s + 2·17-s + 4·19-s − 21-s + 9·23-s − 25-s + 5·27-s + 5·31-s + 5·33-s − 2·35-s + 3·37-s − 5·41-s − 4·43-s + 4·45-s + 13·47-s + 49-s − 2·51-s − 14·53-s + 10·55-s − 4·57-s + 6·59-s + 13·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 0.377·7-s − 2/3·9-s − 1.50·11-s + 0.516·15-s + 0.485·17-s + 0.917·19-s − 0.218·21-s + 1.87·23-s − 1/5·25-s + 0.962·27-s + 0.898·31-s + 0.870·33-s − 0.338·35-s + 0.493·37-s − 0.780·41-s − 0.609·43-s + 0.596·45-s + 1.89·47-s + 1/7·49-s − 0.280·51-s − 1.92·53-s + 1.34·55-s − 0.529·57-s + 0.781·59-s + 1.66·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.434385102\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.434385102\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 13 T + p T^{2} \) | 1.47.an |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 13 T + p T^{2} \) | 1.61.an |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 79 | \( 1 - 15 T + p T^{2} \) | 1.79.ap |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−13.90541906444784, −13.62289293392512, −12.99369090975647, −12.45813981111550, −11.97105538684395, −11.54816579861896, −11.07679478752584, −10.74901142331811, −10.16243809008082, −9.565014939473029, −8.906762782948216, −8.268309589116438, −7.964151970573701, −7.479063993228093, −6.900876362945307, −6.285291835398984, −5.418332056170697, −5.256030530160023, −4.788487950555775, −4.003274148584850, −3.169026927151689, −2.920725269259789, −2.107861208035370, −0.9459741482668573, −0.5197876530363323,
0.5197876530363323, 0.9459741482668573, 2.107861208035370, 2.920725269259789, 3.169026927151689, 4.003274148584850, 4.788487950555775, 5.256030530160023, 5.418332056170697, 6.285291835398984, 6.900876362945307, 7.479063993228093, 7.964151970573701, 8.268309589116438, 8.906762782948216, 9.565014939473029, 10.16243809008082, 10.74901142331811, 11.07679478752584, 11.54816579861896, 11.97105538684395, 12.45813981111550, 12.99369090975647, 13.62289293392512, 13.90541906444784
