| L(s) = 1 | + 3·7-s + 11-s − 13-s − 17-s − 2·19-s + 3·23-s − 2·29-s + 6·31-s + 11·37-s + 5·41-s − 4·43-s + 10·47-s + 2·49-s − 11·53-s − 8·59-s − 13·61-s − 12·67-s − 5·71-s − 10·73-s + 3·77-s + 3·79-s − 12·83-s + 15·89-s − 3·91-s − 17·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 1.13·7-s + 0.301·11-s − 0.277·13-s − 0.242·17-s − 0.458·19-s + 0.625·23-s − 0.371·29-s + 1.07·31-s + 1.80·37-s + 0.780·41-s − 0.609·43-s + 1.45·47-s + 2/7·49-s − 1.51·53-s − 1.04·59-s − 1.66·61-s − 1.46·67-s − 0.593·71-s − 1.17·73-s + 0.341·77-s + 0.337·79-s − 1.31·83-s + 1.58·89-s − 0.314·91-s − 1.72·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 11 T + p T^{2} \) | 1.37.al |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 + 11 T + p T^{2} \) | 1.53.l |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 3 T + p T^{2} \) | 1.79.ad |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
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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
−13.52594599048516, −12.81637751989468, −12.41470006481212, −11.91634584586882, −11.42471842085222, −11.03872275406025, −10.67890431320966, −10.09440734304527, −9.497927591678818, −9.049927840341982, −8.669398239266705, −7.916677416299414, −7.780887353746188, −7.190394188844279, −6.581268903488981, −5.951229201995170, −5.719369797441688, −4.724819842083449, −4.554683948048631, −4.226662390902055, −3.197892808690672, −2.819629835644293, −2.092625973264109, −1.493978259323816, −0.9500097860110219, 0,
0.9500097860110219, 1.493978259323816, 2.092625973264109, 2.819629835644293, 3.197892808690672, 4.226662390902055, 4.554683948048631, 4.724819842083449, 5.719369797441688, 5.951229201995170, 6.581268903488981, 7.190394188844279, 7.780887353746188, 7.916677416299414, 8.669398239266705, 9.049927840341982, 9.497927591678818, 10.09440734304527, 10.67890431320966, 11.03872275406025, 11.42471842085222, 11.91634584586882, 12.41470006481212, 12.81637751989468, 13.52594599048516
