| L(s) = 1 | + 2·2-s + 2·4-s + 3·5-s − 7-s + 6·10-s − 3·11-s − 2·14-s − 4·16-s + 3·17-s + 6·20-s − 6·22-s + 8·23-s + 4·25-s − 2·28-s + 2·29-s + 2·31-s − 8·32-s + 6·34-s − 3·35-s + 4·37-s + 6·41-s − 5·43-s − 6·44-s + 16·46-s − 13·47-s − 6·49-s + 8·50-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 1.34·5-s − 0.377·7-s + 1.89·10-s − 0.904·11-s − 0.534·14-s − 16-s + 0.727·17-s + 1.34·20-s − 1.27·22-s + 1.66·23-s + 4/5·25-s − 0.377·28-s + 0.371·29-s + 0.359·31-s − 1.41·32-s + 1.02·34-s − 0.507·35-s + 0.657·37-s + 0.937·41-s − 0.762·43-s − 0.904·44-s + 2.35·46-s − 1.89·47-s − 6/7·49-s + 1.13·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 172197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 172197 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 19 | \( 1 \) | |
| 53 | \( 1 + T \) | |
| good | 2 | \( 1 - p T + p T^{2} \) | 1.2.ac |
| 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + 13 T + p T^{2} \) | 1.47.n |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−13.42223122118067, −13.10696297076631, −12.71129584691387, −12.28204937350916, −11.72239762816537, −11.05454073605804, −10.77489967091374, −10.11301150319105, −9.691411141981739, −9.245978955449026, −8.837971346965204, −7.913837234348711, −7.685742402925544, −6.718910586637611, −6.423559246222616, −6.131944593165820, −5.344515348388946, −5.121652155789820, −4.786249371695473, −3.993933246427225, −3.215807341841142, −2.955724638616440, −2.475912030468200, −1.740993478686818, −1.057927937842006, 0,
1.057927937842006, 1.740993478686818, 2.475912030468200, 2.955724638616440, 3.215807341841142, 3.993933246427225, 4.786249371695473, 5.121652155789820, 5.344515348388946, 6.131944593165820, 6.423559246222616, 6.718910586637611, 7.685742402925544, 7.913837234348711, 8.837971346965204, 9.245978955449026, 9.691411141981739, 10.11301150319105, 10.77489967091374, 11.05454073605804, 11.72239762816537, 12.28204937350916, 12.71129584691387, 13.10696297076631, 13.42223122118067
