| L(s) = 1 | − 2·5-s − 2·11-s + 2·13-s − 4·17-s − 23-s − 25-s + 10·29-s − 4·37-s − 6·41-s + 4·43-s − 8·47-s + 6·53-s + 4·55-s − 4·59-s − 8·61-s − 4·65-s + 4·67-s + 8·71-s − 6·73-s − 6·79-s + 6·83-s + 8·85-s − 4·89-s − 10·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.603·11-s + 0.554·13-s − 0.970·17-s − 0.208·23-s − 1/5·25-s + 1.85·29-s − 0.657·37-s − 0.937·41-s + 0.609·43-s − 1.16·47-s + 0.824·53-s + 0.539·55-s − 0.520·59-s − 1.02·61-s − 0.496·65-s + 0.488·67-s + 0.949·71-s − 0.702·73-s − 0.675·79-s + 0.658·83-s + 0.867·85-s − 0.423·89-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-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.c |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−13.48025215087067, −13.08438950670044, −12.44114608398298, −12.06566297431681, −11.67424180137156, −11.01357994555112, −10.85178332620712, −10.12932728972238, −9.852798359611073, −9.004630186237546, −8.635307554100796, −8.191700086264345, −7.813451532116483, −7.186282643685190, −6.660107671707558, −6.291699887017038, −5.571192894677454, −5.014095334461881, −4.435188985322584, −4.095389346728493, −3.340082395357583, −2.947044760266630, −2.202669667961932, −1.547792371258727, −0.6752160042391586, 0,
0.6752160042391586, 1.547792371258727, 2.202669667961932, 2.947044760266630, 3.340082395357583, 4.095389346728493, 4.435188985322584, 5.014095334461881, 5.571192894677454, 6.291699887017038, 6.660107671707558, 7.186282643685190, 7.813451532116483, 8.191700086264345, 8.635307554100796, 9.004630186237546, 9.852798359611073, 10.12932728972238, 10.85178332620712, 11.01357994555112, 11.67424180137156, 12.06566297431681, 12.44114608398298, 13.08438950670044, 13.48025215087067
