| L(s) = 1 | − 3-s + 5-s − 7-s + 9-s + 5·11-s − 15-s + 2·17-s + 19-s + 21-s − 6·23-s + 25-s − 27-s + 6·29-s + 31-s − 5·33-s − 35-s − 5·41-s + 4·43-s + 45-s − 3·47-s + 49-s − 2·51-s − 6·53-s + 5·55-s − 57-s − 8·59-s − 12·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.50·11-s − 0.258·15-s + 0.485·17-s + 0.229·19-s + 0.218·21-s − 1.25·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.179·31-s − 0.870·33-s − 0.169·35-s − 0.780·41-s + 0.609·43-s + 0.149·45-s − 0.437·47-s + 1/7·49-s − 0.280·51-s − 0.824·53-s + 0.674·55-s − 0.132·57-s − 1.04·59-s − 1.53·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 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
−14.23978620904661, −13.92335115582706, −13.50137151025344, −12.68970127002295, −12.31859177148486, −11.91542164094512, −11.50603317796210, −10.82534591215412, −10.29261454408888, −9.850403700739159, −9.374080835640635, −8.944175826688751, −8.202365047642191, −7.702586650549243, −6.965436424477312, −6.433401918942716, −6.176220579330502, −5.620645650352412, −4.834470796868922, −4.394732496759460, −3.663926703652080, −3.186806277560413, −2.287506593241149, −1.524173976073768, −1.016483258856973, 0,
1.016483258856973, 1.524173976073768, 2.287506593241149, 3.186806277560413, 3.663926703652080, 4.394732496759460, 4.834470796868922, 5.620645650352412, 6.176220579330502, 6.433401918942716, 6.965436424477312, 7.702586650549243, 8.202365047642191, 8.944175826688751, 9.374080835640635, 9.850403700739159, 10.29261454408888, 10.82534591215412, 11.50603317796210, 11.91542164094512, 12.31859177148486, 12.68970127002295, 13.50137151025344, 13.92335115582706, 14.23978620904661
