| L(s) = 1 | − 3·5-s − 7-s − 11-s + 3·13-s − 17-s + 6·19-s − 2·23-s + 4·25-s − 6·29-s + 3·35-s + 3·37-s − 11·43-s + 49-s + 9·53-s + 3·55-s − 4·59-s − 14·61-s − 9·65-s + 7·67-s + 12·71-s − 73-s + 77-s + 5·79-s + 3·83-s + 3·85-s + 89-s − 3·91-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 0.377·7-s − 0.301·11-s + 0.832·13-s − 0.242·17-s + 1.37·19-s − 0.417·23-s + 4/5·25-s − 1.11·29-s + 0.507·35-s + 0.493·37-s − 1.67·43-s + 1/7·49-s + 1.23·53-s + 0.404·55-s − 0.520·59-s − 1.79·61-s − 1.11·65-s + 0.855·67-s + 1.42·71-s − 0.117·73-s + 0.113·77-s + 0.562·79-s + 0.329·83-s + 0.325·85-s + 0.105·89-s − 0.314·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 17 | \( 1 + T \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 11 T + p T^{2} \) | 1.43.l |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 79 | \( 1 - 5 T + p T^{2} \) | 1.79.af |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 - T + p T^{2} \) | 1.89.ab |
| 97 | \( 1 + 13 T + p T^{2} \) | 1.97.n |
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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
−16.03072504754709, −15.61215787336212, −15.22862051138629, −14.64638411704277, −13.75795899612453, −13.49510942203292, −12.78428444531137, −12.15818069509999, −11.69394480852889, −11.20427557402586, −10.71059849276735, −9.906934633829579, −9.378004033368456, −8.667588510610227, −8.074524889984497, −7.605712140431646, −7.053416892479365, −6.323705229953880, −5.609480018910547, −4.922451315684561, −4.129065603611591, −3.529732516503724, −3.119902147691906, −1.999065856410795, −0.9332406389424395, 0,
0.9332406389424395, 1.999065856410795, 3.119902147691906, 3.529732516503724, 4.129065603611591, 4.922451315684561, 5.609480018910547, 6.323705229953880, 7.053416892479365, 7.605712140431646, 8.074524889984497, 8.667588510610227, 9.378004033368456, 9.906934633829579, 10.71059849276735, 11.20427557402586, 11.69394480852889, 12.15818069509999, 12.78428444531137, 13.49510942203292, 13.75795899612453, 14.64638411704277, 15.22862051138629, 15.61215787336212, 16.03072504754709
