| L(s) = 1 | − 2·4-s + 2·13-s + 4·16-s − 7·19-s − 5·25-s + 11·31-s − 10·37-s − 13·43-s − 4·52-s − 13·61-s − 8·64-s − 16·67-s − 7·73-s + 14·76-s − 4·79-s + 5·97-s + 10·100-s + 20·103-s − 19·109-s + ⋯ |
| L(s) = 1 | − 4-s + 0.554·13-s + 16-s − 1.60·19-s − 25-s + 1.97·31-s − 1.64·37-s − 1.98·43-s − 0.554·52-s − 1.66·61-s − 64-s − 1.95·67-s − 0.819·73-s + 1.60·76-s − 0.450·79-s + 0.507·97-s + 100-s + 1.97·103-s − 1.81·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 11 T + p T^{2} \) | 1.31.al |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 13 T + p T^{2} \) | 1.43.n |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 + 16 T + p T^{2} \) | 1.67.q |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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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
−9.066839472611968948838384775491, −8.509905602511589997008412828041, −7.86092323383857706747859641278, −6.62218489901874158644992085419, −5.89971919728528192572309310294, −4.82508280591616755086822235661, −4.14835908916201458718626857057, −3.15120267703981185726680892993, −1.63769139620368418229574505514, 0,
1.63769139620368418229574505514, 3.15120267703981185726680892993, 4.14835908916201458718626857057, 4.82508280591616755086822235661, 5.89971919728528192572309310294, 6.62218489901874158644992085419, 7.86092323383857706747859641278, 8.509905602511589997008412828041, 9.066839472611968948838384775491
