| L(s) = 1 | + 2·5-s + 4·13-s − 4·17-s − 2·19-s − 23-s − 25-s + 6·29-s + 2·31-s − 6·37-s − 2·41-s − 4·43-s − 6·47-s − 6·53-s + 2·61-s + 8·65-s + 4·67-s + 12·71-s + 10·73-s + 8·79-s + 6·83-s − 8·85-s − 4·95-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.10·13-s − 0.970·17-s − 0.458·19-s − 0.208·23-s − 1/5·25-s + 1.11·29-s + 0.359·31-s − 0.986·37-s − 0.312·41-s − 0.609·43-s − 0.875·47-s − 0.824·53-s + 0.256·61-s + 0.992·65-s + 0.488·67-s + 1.42·71-s + 1.17·73-s + 0.900·79-s + 0.658·83-s − 0.867·85-s − 0.410·95-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−13.59054760764318, −13.19913966321204, −12.55600180179185, −12.16282624778114, −11.55996229555840, −11.01038286895606, −10.69707897867049, −10.19693915150091, −9.580776273848482, −9.350082290728400, −8.562832299550974, −8.338802750532948, −7.884361303822796, −6.901802005116620, −6.577350961389172, −6.300398444500792, −5.659897613229641, −5.074881386277726, −4.650516135498243, −3.891170871403522, −3.477813226000600, −2.714547817621047, −2.097147079430071, −1.653712780998674, −0.9189288831674399, 0,
0.9189288831674399, 1.653712780998674, 2.097147079430071, 2.714547817621047, 3.477813226000600, 3.891170871403522, 4.650516135498243, 5.074881386277726, 5.659897613229641, 6.300398444500792, 6.577350961389172, 6.901802005116620, 7.884361303822796, 8.338802750532948, 8.562832299550974, 9.350082290728400, 9.580776273848482, 10.19693915150091, 10.69707897867049, 11.01038286895606, 11.55996229555840, 12.16282624778114, 12.55600180179185, 13.19913966321204, 13.59054760764318
