| L(s) = 1 | − 4·7-s − 3·9-s − 11-s + 2·17-s − 4·19-s − 4·23-s − 5·25-s + 6·29-s − 4·31-s − 8·37-s − 12·41-s − 12·43-s + 4·47-s + 9·49-s − 6·53-s + 4·59-s + 10·61-s + 12·63-s − 4·67-s − 4·71-s + 4·73-s + 4·77-s + 8·79-s + 9·81-s + 12·83-s − 16·89-s − 8·97-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 9-s − 0.301·11-s + 0.485·17-s − 0.917·19-s − 0.834·23-s − 25-s + 1.11·29-s − 0.718·31-s − 1.31·37-s − 1.87·41-s − 1.82·43-s + 0.583·47-s + 9/7·49-s − 0.824·53-s + 0.520·59-s + 1.28·61-s + 1.51·63-s − 0.488·67-s − 0.474·71-s + 0.468·73-s + 0.455·77-s + 0.900·79-s + 81-s + 1.31·83-s − 1.69·89-s − 0.812·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 118976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 118976 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−13.75567474034389, −13.36532425796429, −12.89061424063065, −12.30341063159317, −11.93288356231501, −11.59136304664614, −10.72157687412185, −10.41924737109763, −9.836630769349838, −9.630856834838024, −8.800085934775292, −8.419166719825913, −8.097102214612651, −7.251653377893592, −6.710273587559222, −6.379573539026486, −5.789293721196665, −5.366864335003756, −4.724765553841681, −3.896518580017002, −3.323714104202645, −3.161446777507605, −2.218561383898165, −1.803090975339164, −0.5249714678703789, 0,
0.5249714678703789, 1.803090975339164, 2.218561383898165, 3.161446777507605, 3.323714104202645, 3.896518580017002, 4.724765553841681, 5.366864335003756, 5.789293721196665, 6.379573539026486, 6.710273587559222, 7.251653377893592, 8.097102214612651, 8.419166719825913, 8.800085934775292, 9.630856834838024, 9.836630769349838, 10.41924737109763, 10.72157687412185, 11.59136304664614, 11.93288356231501, 12.30341063159317, 12.89061424063065, 13.36532425796429, 13.75567474034389
