| L(s) = 1 | + 3·7-s − 4·11-s + 13-s + 2·17-s − 7·19-s − 3·23-s − 2·29-s + 2·31-s + 6·37-s − 7·41-s + 8·43-s − 9·47-s + 2·49-s − 11·53-s − 7·59-s + 14·61-s + 10·67-s + 6·71-s − 8·73-s − 12·77-s + 10·79-s − 2·83-s − 6·89-s + 3·91-s − 4·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 1.13·7-s − 1.20·11-s + 0.277·13-s + 0.485·17-s − 1.60·19-s − 0.625·23-s − 0.371·29-s + 0.359·31-s + 0.986·37-s − 1.09·41-s + 1.21·43-s − 1.31·47-s + 2/7·49-s − 1.51·53-s − 0.911·59-s + 1.79·61-s + 1.22·67-s + 0.712·71-s − 0.936·73-s − 1.36·77-s + 1.12·79-s − 0.219·83-s − 0.635·89-s + 0.314·91-s − 0.406·97-s + 0.0995·101-s + 0.0985·103-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 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 7 T + p T^{2} \) | 1.41.h |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 + 11 T + p T^{2} \) | 1.53.l |
| 59 | \( 1 + 7 T + p T^{2} \) | 1.59.h |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
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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
−12.95585627131065, −12.61816702193479, −12.25777272560894, −11.46560326095241, −11.14413485746080, −10.93255847560378, −10.24293180331833, −9.949667808436981, −9.389122231356711, −8.680105855310710, −8.239650941171815, −7.990076205293728, −7.662508189622625, −6.875955756804029, −6.416205894609835, −5.871268967391499, −5.345196598857432, −4.893917742515909, −4.413110221728843, −3.933203246423419, −3.218432315442507, −2.605116943266198, −2.013507951878811, −1.636564945496238, −0.7493458925384838, 0,
0.7493458925384838, 1.636564945496238, 2.013507951878811, 2.605116943266198, 3.218432315442507, 3.933203246423419, 4.413110221728843, 4.893917742515909, 5.345196598857432, 5.871268967391499, 6.416205894609835, 6.875955756804029, 7.662508189622625, 7.990076205293728, 8.239650941171815, 8.680105855310710, 9.389122231356711, 9.949667808436981, 10.24293180331833, 10.93255847560378, 11.14413485746080, 11.46560326095241, 12.25777272560894, 12.61816702193479, 12.95585627131065
