| L(s) = 1 | − 5-s − 3·7-s − 3·9-s + 3·11-s − 4·17-s − 7·19-s − 4·23-s + 25-s + 8·29-s − 10·31-s + 3·35-s − 3·37-s − 2·41-s − 6·43-s + 3·45-s − 47-s + 2·49-s + 9·53-s − 3·55-s + 4·59-s + 14·61-s + 9·63-s − 4·67-s + 6·71-s + 4·73-s − 9·77-s + 10·79-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.13·7-s − 9-s + 0.904·11-s − 0.970·17-s − 1.60·19-s − 0.834·23-s + 1/5·25-s + 1.48·29-s − 1.79·31-s + 0.507·35-s − 0.493·37-s − 0.312·41-s − 0.914·43-s + 0.447·45-s − 0.145·47-s + 2/7·49-s + 1.23·53-s − 0.404·55-s + 0.520·59-s + 1.79·61-s + 1.13·63-s − 0.488·67-s + 0.712·71-s + 0.468·73-s − 1.02·77-s + 1.12·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - T + p T^{2} \) | 1.89.ab |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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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
−14.59477800072603, −14.32311014031272, −13.62212228391618, −13.11189260013583, −12.67977368834449, −12.09433511605143, −11.67600112042171, −11.16986227515750, −10.53792747758207, −10.15607797219728, −9.391160326460071, −8.928463496476883, −8.495674062888970, −8.112630788684683, −7.095499916087887, −6.683178583919961, −6.348117221848471, −5.746410987222287, −4.993155455441237, −4.281549714817078, −3.678062992770176, −3.343102825262914, −2.367672384779636, −1.978029075079015, −0.6552508418354983, 0,
0.6552508418354983, 1.978029075079015, 2.367672384779636, 3.343102825262914, 3.678062992770176, 4.281549714817078, 4.993155455441237, 5.746410987222287, 6.348117221848471, 6.683178583919961, 7.095499916087887, 8.112630788684683, 8.495674062888970, 8.928463496476883, 9.391160326460071, 10.15607797219728, 10.53792747758207, 11.16986227515750, 11.67600112042171, 12.09433511605143, 12.67977368834449, 13.11189260013583, 13.62212228391618, 14.32311014031272, 14.59477800072603
