| L(s) = 1 | − 2-s + 4-s − 7-s − 8-s + 11-s − 2·13-s + 14-s + 16-s − 6·17-s − 4·19-s − 22-s + 2·26-s − 28-s + 6·29-s − 4·31-s − 32-s + 6·34-s − 2·37-s + 4·38-s + 6·41-s + 4·43-s + 44-s + 49-s − 2·52-s − 6·53-s + 56-s − 6·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 0.301·11-s − 0.554·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.917·19-s − 0.213·22-s + 0.392·26-s − 0.188·28-s + 1.11·29-s − 0.718·31-s − 0.176·32-s + 1.02·34-s − 0.328·37-s + 0.648·38-s + 0.937·41-s + 0.609·43-s + 0.150·44-s + 1/7·49-s − 0.277·52-s − 0.824·53-s + 0.133·56-s − 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 34650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| good | 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 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.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 16 T + p T^{2} \) | 1.67.aq |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−15.45332092037252, −14.56623618261216, −14.44185404919445, −13.60944822985777, −13.01065333333972, −12.56227540417428, −12.09160221568897, −11.33439788822069, −10.94627978323779, −10.46445255589365, −9.792936316418154, −9.330509590667722, −8.797841044333516, −8.355904455766881, −7.652503152587784, −7.032362131761644, −6.530158329971293, −6.136995958090974, −5.266863095657642, −4.538349514138089, −4.004908310586306, −3.143982266618892, −2.378912321187591, −1.934805910472861, −0.8290224835746324, 0,
0.8290224835746324, 1.934805910472861, 2.378912321187591, 3.143982266618892, 4.004908310586306, 4.538349514138089, 5.266863095657642, 6.136995958090974, 6.530158329971293, 7.032362131761644, 7.652503152587784, 8.355904455766881, 8.797841044333516, 9.330509590667722, 9.792936316418154, 10.46445255589365, 10.94627978323779, 11.33439788822069, 12.09160221568897, 12.56227540417428, 13.01065333333972, 13.60944822985777, 14.44185404919445, 14.56623618261216, 15.45332092037252
