| L(s) = 1 | + 2·5-s − 4·7-s + 13-s + 2·17-s + 6·19-s − 8·23-s − 25-s + 4·29-s − 4·31-s − 8·35-s − 6·37-s + 8·43-s + 9·49-s − 8·59-s + 10·61-s + 2·65-s + 14·67-s − 8·71-s − 2·73-s − 14·79-s − 4·83-s + 4·85-s − 8·89-s − 4·91-s + 12·95-s − 10·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.51·7-s + 0.277·13-s + 0.485·17-s + 1.37·19-s − 1.66·23-s − 1/5·25-s + 0.742·29-s − 0.718·31-s − 1.35·35-s − 0.986·37-s + 1.21·43-s + 9/7·49-s − 1.04·59-s + 1.28·61-s + 0.248·65-s + 1.71·67-s − 0.949·71-s − 0.234·73-s − 1.57·79-s − 0.439·83-s + 0.433·85-s − 0.847·89-s − 0.419·91-s + 1.23·95-s − 1.01·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14976 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−16.24652362379902, −15.79336557753049, −15.56373651997060, −14.37199999494938, −14.01391624352627, −13.71107361586432, −12.88706111407259, −12.59940911234460, −11.90599364179974, −11.35195139684243, −10.23775938691545, −10.17694271613083, −9.546735118898554, −9.118829217963225, −8.315693683306055, −7.545394224652614, −6.939873195857850, −6.250197975991835, −5.760145199039189, −5.344086990604997, −4.183788616310920, −3.533998865929293, −2.908094303586273, −2.101421022312421, −1.156097311396371, 0,
1.156097311396371, 2.101421022312421, 2.908094303586273, 3.533998865929293, 4.183788616310920, 5.344086990604997, 5.760145199039189, 6.250197975991835, 6.939873195857850, 7.545394224652614, 8.315693683306055, 9.118829217963225, 9.546735118898554, 10.17694271613083, 10.23775938691545, 11.35195139684243, 11.90599364179974, 12.59940911234460, 12.88706111407259, 13.71107361586432, 14.01391624352627, 14.37199999494938, 15.56373651997060, 15.79336557753049, 16.24652362379902
