| L(s) = 1 | − 2·7-s + 4·11-s − 13-s − 5·17-s − 2·19-s + 4·23-s + 3·29-s + 6·31-s − 5·37-s − 6·41-s − 2·47-s − 3·49-s − 6·53-s + 8·59-s + 13·61-s + 10·67-s + 6·71-s − 11·73-s − 8·77-s + 4·79-s − 6·83-s − 7·89-s + 2·91-s + 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 1.20·11-s − 0.277·13-s − 1.21·17-s − 0.458·19-s + 0.834·23-s + 0.557·29-s + 1.07·31-s − 0.821·37-s − 0.937·41-s − 0.291·47-s − 3/7·49-s − 0.824·53-s + 1.04·59-s + 1.66·61-s + 1.22·67-s + 0.712·71-s − 1.28·73-s − 0.911·77-s + 0.450·79-s − 0.658·83-s − 0.741·89-s + 0.209·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 13 T + p T^{2} \) | 1.61.an |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 7 T + p T^{2} \) | 1.89.h |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.08564501355335, −12.61874105863737, −12.12104255881023, −11.66720000106313, −11.25899617205825, −10.81331273274843, −10.13318464259703, −9.849717540296152, −9.347699504872402, −8.875224834848054, −8.415565330719606, −8.120401431975466, −7.090334061204450, −6.835955719980303, −6.598149645848441, −6.079809062077247, −5.370831769643919, −4.811085193190214, −4.369342888529255, −3.783964784309769, −3.314477074134903, −2.690027267070277, −2.132791215803079, −1.449753949260759, −0.7489631705713965, 0,
0.7489631705713965, 1.449753949260759, 2.132791215803079, 2.690027267070277, 3.314477074134903, 3.783964784309769, 4.369342888529255, 4.811085193190214, 5.370831769643919, 6.079809062077247, 6.598149645848441, 6.835955719980303, 7.090334061204450, 8.120401431975466, 8.415565330719606, 8.875224834848054, 9.347699504872402, 9.849717540296152, 10.13318464259703, 10.81331273274843, 11.25899617205825, 11.66720000106313, 12.12104255881023, 12.61874105863737, 13.08564501355335
