| L(s) = 1 | − 2·2-s + 2·4-s + 3·7-s − 2·11-s + 13-s − 6·14-s − 4·16-s + 2·17-s + 4·22-s + 6·23-s − 2·26-s + 6·28-s + 10·29-s + 3·31-s + 8·32-s − 4·34-s + 2·37-s − 8·41-s − 43-s − 4·44-s − 12·46-s + 2·47-s + 2·49-s + 2·52-s + 4·53-s − 20·58-s − 10·59-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 1.13·7-s − 0.603·11-s + 0.277·13-s − 1.60·14-s − 16-s + 0.485·17-s + 0.852·22-s + 1.25·23-s − 0.392·26-s + 1.13·28-s + 1.85·29-s + 0.538·31-s + 1.41·32-s − 0.685·34-s + 0.328·37-s − 1.24·41-s − 0.152·43-s − 0.603·44-s − 1.76·46-s + 0.291·47-s + 2/7·49-s + 0.277·52-s + 0.549·53-s − 2.62·58-s − 1.30·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 19 | \( 1 \) | |
| good | 2 | \( 1 + p T + p T^{2} \) | 1.2.c |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 17 T + p T^{2} \) | 1.97.ar |
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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
−14.29302020525419, −13.60982803056712, −13.44718793161925, −12.63474622751841, −11.98076803429965, −11.64655442816053, −11.01985665864035, −10.60132470858617, −10.30038250556009, −9.692807668000407, −9.134070472118813, −8.533621660699760, −8.282337456627431, −7.820664213768997, −7.308148034092651, −6.713889798636367, −6.208462664638209, −5.235594497481326, −4.927646414383228, −4.411996709919899, −3.491996875584040, −2.715718006448781, −2.204966016076150, −1.164312159815478, −1.147454454079549, 0,
1.147454454079549, 1.164312159815478, 2.204966016076150, 2.715718006448781, 3.491996875584040, 4.411996709919899, 4.927646414383228, 5.235594497481326, 6.208462664638209, 6.713889798636367, 7.308148034092651, 7.820664213768997, 8.282337456627431, 8.533621660699760, 9.134070472118813, 9.692807668000407, 10.30038250556009, 10.60132470858617, 11.01985665864035, 11.64655442816053, 11.98076803429965, 12.63474622751841, 13.44718793161925, 13.60982803056712, 14.29302020525419
