| L(s) = 1 | − 4·7-s + 4·11-s − 13-s + 4·17-s − 6·19-s − 5·25-s + 6·29-s + 4·31-s − 6·37-s − 6·41-s + 4·43-s + 8·47-s + 9·49-s − 6·53-s + 12·59-s − 10·61-s + 2·67-s + 12·71-s − 10·73-s − 16·77-s − 10·79-s + 4·83-s + 10·89-s + 4·91-s + 14·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 1.20·11-s − 0.277·13-s + 0.970·17-s − 1.37·19-s − 25-s + 1.11·29-s + 0.718·31-s − 0.986·37-s − 0.937·41-s + 0.609·43-s + 1.16·47-s + 9/7·49-s − 0.824·53-s + 1.56·59-s − 1.28·61-s + 0.244·67-s + 1.42·71-s − 1.17·73-s − 1.82·77-s − 1.12·79-s + 0.439·83-s + 1.05·89-s + 0.419·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14976 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 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
−16.36854179028709, −15.73802361730217, −15.36596912387369, −14.62844591073120, −14.05932465141463, −13.62909902905049, −12.89346984264113, −12.33354283141115, −12.05933323586496, −11.37454143673116, −10.38158889446808, −10.14702269166543, −9.539527318092646, −8.919674007886505, −8.416638684404468, −7.548370695915301, −6.866354456883366, −6.327827638303176, −5.995757904696912, −5.032671341527104, −4.139902637467027, −3.661408841386051, −2.953185035081968, −2.110337569430104, −1.048210935708929, 0,
1.048210935708929, 2.110337569430104, 2.953185035081968, 3.661408841386051, 4.139902637467027, 5.032671341527104, 5.995757904696912, 6.327827638303176, 6.866354456883366, 7.548370695915301, 8.416638684404468, 8.919674007886505, 9.539527318092646, 10.14702269166543, 10.38158889446808, 11.37454143673116, 12.05933323586496, 12.33354283141115, 12.89346984264113, 13.62909902905049, 14.05932465141463, 14.62844591073120, 15.36596912387369, 15.73802361730217, 16.36854179028709
