| L(s) = 1 | − 4·7-s − 3·11-s − 4·13-s − 3·17-s − 4·19-s + 6·23-s − 6·29-s − 8·31-s + 8·37-s + 6·41-s + 43-s − 12·47-s + 9·49-s + 9·59-s − 8·61-s + 4·67-s − 6·71-s − 14·73-s + 12·77-s − 8·79-s − 9·83-s + 9·89-s + 16·91-s + 7·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 0.904·11-s − 1.10·13-s − 0.727·17-s − 0.917·19-s + 1.25·23-s − 1.11·29-s − 1.43·31-s + 1.31·37-s + 0.937·41-s + 0.152·43-s − 1.75·47-s + 9/7·49-s + 1.17·59-s − 1.02·61-s + 0.488·67-s − 0.712·71-s − 1.63·73-s + 1.36·77-s − 0.900·79-s − 0.987·83-s + 0.953·89-s + 1.67·91-s + 0.710·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−13.35582890224136, −13.11167964837975, −12.84032937998952, −12.65257971994275, −11.77339871373245, −11.22932118140975, −10.89877331579179, −10.16621107012997, −9.972822118121600, −9.242320968118900, −9.079312810634582, −8.449811415216074, −7.558452518763458, −7.398719111589713, −6.846019796644321, −6.219422760503557, −5.848138530832417, −5.168663560224891, −4.646696773214769, −4.062727419244147, −3.362250046683918, −2.839544139583022, −2.396132131887028, −1.719601999274338, −0.5225622868916381, 0,
0.5225622868916381, 1.719601999274338, 2.396132131887028, 2.839544139583022, 3.362250046683918, 4.062727419244147, 4.646696773214769, 5.168663560224891, 5.848138530832417, 6.219422760503557, 6.846019796644321, 7.398719111589713, 7.558452518763458, 8.449811415216074, 9.079312810634582, 9.242320968118900, 9.972822118121600, 10.16621107012997, 10.89877331579179, 11.22932118140975, 11.77339871373245, 12.65257971994275, 12.84032937998952, 13.11167964837975, 13.35582890224136
