| L(s) = 1 | − 3-s − 4·5-s + 9-s + 4·15-s − 6·17-s + 4·19-s − 6·23-s + 11·25-s − 27-s − 6·29-s + 6·37-s − 10·41-s + 8·43-s − 4·45-s − 6·47-s + 6·51-s − 12·53-s − 4·57-s + 8·59-s + 4·61-s − 12·67-s + 6·69-s + 10·71-s + 2·73-s − 11·75-s − 2·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.78·5-s + 1/3·9-s + 1.03·15-s − 1.45·17-s + 0.917·19-s − 1.25·23-s + 11/5·25-s − 0.192·27-s − 1.11·29-s + 0.986·37-s − 1.56·41-s + 1.21·43-s − 0.596·45-s − 0.875·47-s + 0.840·51-s − 1.64·53-s − 0.529·57-s + 1.04·59-s + 0.512·61-s − 1.46·67-s + 0.722·69-s + 1.18·71-s + 0.234·73-s − 1.27·75-s − 0.225·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 142296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1149979919\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1149979919\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−13.35712679656602, −12.80977569626404, −12.19270715136200, −12.03189839850991, −11.37516580481365, −11.14166997561828, −10.84016654460154, −10.03450342476113, −9.566680780013498, −8.995618579265645, −8.429410084243384, −7.827705911857130, −7.706814683635712, −6.960880580664148, −6.615599495206942, −6.016481963942428, −5.279061950643896, −4.814652267638294, −4.232190590216180, −3.867294937547995, −3.348328506976311, −2.595346492310976, −1.831791651214703, −1.006564952172716, −0.1223681354680014,
0.1223681354680014, 1.006564952172716, 1.831791651214703, 2.595346492310976, 3.348328506976311, 3.867294937547995, 4.232190590216180, 4.814652267638294, 5.279061950643896, 6.016481963942428, 6.615599495206942, 6.960880580664148, 7.706814683635712, 7.827705911857130, 8.429410084243384, 8.995618579265645, 9.566680780013498, 10.03450342476113, 10.84016654460154, 11.14166997561828, 11.37516580481365, 12.03189839850991, 12.19270715136200, 12.80977569626404, 13.35712679656602
