| L(s) = 1 | − 2·3-s − 2·4-s − 3·5-s + 9-s + 3·11-s + 4·12-s − 3·13-s + 6·15-s + 4·16-s − 6·17-s + 6·20-s + 6·23-s + 4·25-s + 4·27-s + 6·29-s − 3·31-s − 6·33-s − 2·36-s + 6·39-s − 12·43-s − 6·44-s − 3·45-s + 6·47-s − 8·48-s + 12·51-s + 6·52-s + 9·53-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 4-s − 1.34·5-s + 1/3·9-s + 0.904·11-s + 1.15·12-s − 0.832·13-s + 1.54·15-s + 16-s − 1.45·17-s + 1.34·20-s + 1.25·23-s + 4/5·25-s + 0.769·27-s + 1.11·29-s − 0.538·31-s − 1.04·33-s − 1/3·36-s + 0.960·39-s − 1.82·43-s − 0.904·44-s − 0.447·45-s + 0.875·47-s − 1.15·48-s + 1.68·51-s + 0.832·52-s + 1.23·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67081 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67081 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) | |
| 37 | \( 1 \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 - 15 T + p T^{2} \) | 1.97.ap |
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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
−14.58055093087565, −13.89050632350212, −13.32942681086732, −12.86177357329599, −12.19644972533355, −11.95756114201191, −11.54001105684171, −11.02014561525269, −10.48344648055168, −10.01169346884110, −9.176810282499253, −8.748096317495685, −8.529600011806021, −7.551488438673009, −7.269640445415855, −6.511051324302911, −6.216149475444301, −5.170654792117233, −4.937091359336038, −4.451898553632142, −3.899439857086371, −3.309714185389321, −2.484996163367379, −1.302725811550292, −0.5816150709979061, 0,
0.5816150709979061, 1.302725811550292, 2.484996163367379, 3.309714185389321, 3.899439857086371, 4.451898553632142, 4.937091359336038, 5.170654792117233, 6.216149475444301, 6.511051324302911, 7.269640445415855, 7.551488438673009, 8.529600011806021, 8.748096317495685, 9.176810282499253, 10.01169346884110, 10.48344648055168, 11.02014561525269, 11.54001105684171, 11.95756114201191, 12.19644972533355, 12.86177357329599, 13.32942681086732, 13.89050632350212, 14.58055093087565
