| L(s) = 1 | + 4·5-s − 11-s − 6·13-s + 2·19-s − 4·23-s + 11·25-s + 2·29-s + 2·31-s + 2·37-s − 4·43-s + 6·47-s − 2·53-s − 4·55-s − 14·61-s − 24·65-s − 12·67-s − 8·71-s + 4·73-s + 8·79-s + 2·83-s − 14·89-s + 8·95-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 0.301·11-s − 1.66·13-s + 0.458·19-s − 0.834·23-s + 11/5·25-s + 0.371·29-s + 0.359·31-s + 0.328·37-s − 0.609·43-s + 0.875·47-s − 0.274·53-s − 0.539·55-s − 1.79·61-s − 2.97·65-s − 1.46·67-s − 0.949·71-s + 0.468·73-s + 0.900·79-s + 0.219·83-s − 1.48·89-s + 0.820·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 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
−13.69869785101258, −13.13971442352904, −12.63428220631221, −12.06782509939156, −11.93003558071018, −10.98480935556824, −10.52289299252890, −10.14786176346021, −9.638247992118971, −9.493617402674124, −8.852612788441162, −8.285466982991964, −7.585150756080822, −7.247440473100401, −6.614233268666624, −6.024910701581685, −5.766047112628742, −5.068948917427995, −4.769163195680381, −4.161997739615381, −3.031061586355206, −2.852957371479951, −2.094158539712722, −1.770615521888139, −0.9311197575093329, 0,
0.9311197575093329, 1.770615521888139, 2.094158539712722, 2.852957371479951, 3.031061586355206, 4.161997739615381, 4.769163195680381, 5.068948917427995, 5.766047112628742, 6.024910701581685, 6.614233268666624, 7.247440473100401, 7.585150756080822, 8.285466982991964, 8.852612788441162, 9.493617402674124, 9.638247992118971, 10.14786176346021, 10.52289299252890, 10.98480935556824, 11.93003558071018, 12.06782509939156, 12.63428220631221, 13.13971442352904, 13.69869785101258
