| L(s) = 1 | − 2·4-s + 11-s − 2·13-s + 4·16-s + 17-s + 5·19-s − 3·23-s + 5·29-s + 4·31-s − 2·37-s + 2·41-s − 11·43-s − 2·44-s − 10·47-s + 4·52-s + 11·53-s + 7·59-s + 13·61-s − 8·64-s − 8·67-s − 2·68-s + 6·71-s + 14·73-s − 10·76-s + 4·79-s − 9·83-s − 17·89-s + ⋯ |
| L(s) = 1 | − 4-s + 0.301·11-s − 0.554·13-s + 16-s + 0.242·17-s + 1.14·19-s − 0.625·23-s + 0.928·29-s + 0.718·31-s − 0.328·37-s + 0.312·41-s − 1.67·43-s − 0.301·44-s − 1.45·47-s + 0.554·52-s + 1.51·53-s + 0.911·59-s + 1.66·61-s − 64-s − 0.977·67-s − 0.242·68-s + 0.712·71-s + 1.63·73-s − 1.14·76-s + 0.450·79-s − 0.987·83-s − 1.80·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 11 T + p T^{2} \) | 1.43.l |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 - 11 T + p T^{2} \) | 1.53.al |
| 59 | \( 1 - 7 T + p T^{2} \) | 1.59.ah |
| 61 | \( 1 - 13 T + p T^{2} \) | 1.61.an |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 + 17 T + p T^{2} \) | 1.89.r |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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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
−13.83342559048687, −13.41916812971638, −12.84029632932566, −12.37791857841104, −11.85125085394572, −11.58740578830988, −10.88314040719571, −9.989963690551038, −9.920390976615334, −9.687315105314278, −8.668542202400468, −8.568200602753852, −7.987119972280012, −7.416459923707680, −6.837267957645966, −6.316881005947375, −5.603577838621605, −5.085535042683380, −4.828290562223089, −4.001242363753375, −3.635367603739787, −2.975919471492769, −2.312009621826377, −1.392897355048550, −0.8392521784214429, 0,
0.8392521784214429, 1.392897355048550, 2.312009621826377, 2.975919471492769, 3.635367603739787, 4.001242363753375, 4.828290562223089, 5.085535042683380, 5.603577838621605, 6.316881005947375, 6.837267957645966, 7.416459923707680, 7.987119972280012, 8.568200602753852, 8.668542202400468, 9.687315105314278, 9.920390976615334, 9.989963690551038, 10.88314040719571, 11.58740578830988, 11.85125085394572, 12.37791857841104, 12.84029632932566, 13.41916812971638, 13.83342559048687
