| L(s) = 1 | + 2-s + 4-s + 3·7-s + 8-s + 3·11-s + 3·14-s + 16-s − 4·17-s + 19-s + 3·22-s − 8·23-s − 5·25-s + 3·28-s − 4·31-s + 32-s − 4·34-s − 6·37-s + 38-s + 8·41-s + 3·43-s + 3·44-s − 8·46-s + 12·47-s + 2·49-s − 5·50-s + 10·53-s + 3·56-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.13·7-s + 0.353·8-s + 0.904·11-s + 0.801·14-s + 1/4·16-s − 0.970·17-s + 0.229·19-s + 0.639·22-s − 1.66·23-s − 25-s + 0.566·28-s − 0.718·31-s + 0.176·32-s − 0.685·34-s − 0.986·37-s + 0.162·38-s + 1.24·41-s + 0.457·43-s + 0.452·44-s − 1.17·46-s + 1.75·47-s + 2/7·49-s − 0.707·50-s + 1.37·53-s + 0.400·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136242 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 29 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 - 3 T + p T^{2} \) | 1.43.ad |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 9 T + p T^{2} \) | 1.61.aj |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−13.66564629086180, −13.44486958850838, −12.62173658040145, −12.09099956095976, −11.93959678509984, −11.34576096063301, −10.89691374173576, −10.55740864237580, −9.788252121524692, −9.343943436103603, −8.763300885803709, −8.234337156181204, −7.793086889699789, −7.172679099803638, −6.824119415717023, −6.049754781612599, −5.659539390647231, −5.261567124437567, −4.331348668793500, −4.140160567144275, −3.805339612461198, −2.796737764456794, −2.159698583246993, −1.767534329612956, −1.066943217546104, 0,
1.066943217546104, 1.767534329612956, 2.159698583246993, 2.796737764456794, 3.805339612461198, 4.140160567144275, 4.331348668793500, 5.261567124437567, 5.659539390647231, 6.049754781612599, 6.824119415717023, 7.172679099803638, 7.793086889699789, 8.234337156181204, 8.763300885803709, 9.343943436103603, 9.788252121524692, 10.55740864237580, 10.89691374173576, 11.34576096063301, 11.93959678509984, 12.09099956095976, 12.62173658040145, 13.44486958850838, 13.66564629086180
