| L(s) = 1 | − 2·5-s − 4·7-s − 2·13-s + 4·19-s + 4·23-s − 25-s − 31-s + 8·35-s + 2·37-s − 2·41-s − 12·43-s − 10·47-s + 9·49-s + 8·53-s + 14·59-s + 2·61-s + 4·65-s − 4·67-s + 6·71-s + 6·73-s + 16·83-s − 4·89-s + 8·91-s − 8·95-s − 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.51·7-s − 0.554·13-s + 0.917·19-s + 0.834·23-s − 1/5·25-s − 0.179·31-s + 1.35·35-s + 0.328·37-s − 0.312·41-s − 1.82·43-s − 1.45·47-s + 9/7·49-s + 1.09·53-s + 1.82·59-s + 0.256·61-s + 0.496·65-s − 0.488·67-s + 0.712·71-s + 0.702·73-s + 1.75·83-s − 0.423·89-s + 0.838·91-s − 0.820·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 31 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 - 8 T + p T^{2} \) | 1.53.ai |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 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
−16.15171719676577, −15.47615657307051, −15.17504629418115, −14.57328784146833, −13.78205063941318, −13.21551669763687, −12.85167268762541, −12.19649662006642, −11.64192642751285, −11.31012795280651, −10.30980034275442, −9.929212936838786, −9.423070021802047, −8.764310588586740, −8.065778338945997, −7.476637008702057, −6.800369620646159, −6.527828404308517, −5.502570125552049, −5.018428456625688, −4.093446783031433, −3.420357209800297, −3.084685513058756, −2.113788863080806, −0.8367650172161893, 0,
0.8367650172161893, 2.113788863080806, 3.084685513058756, 3.420357209800297, 4.093446783031433, 5.018428456625688, 5.502570125552049, 6.527828404308517, 6.800369620646159, 7.476637008702057, 8.065778338945997, 8.764310588586740, 9.423070021802047, 9.929212936838786, 10.30980034275442, 11.31012795280651, 11.64192642751285, 12.19649662006642, 12.85167268762541, 13.21551669763687, 13.78205063941318, 14.57328784146833, 15.17504629418115, 15.47615657307051, 16.15171719676577
