| L(s) = 1 | − 3·5-s − 2·7-s + 2·11-s − 2·17-s + 5·19-s + 8·23-s + 4·25-s − 8·29-s − 5·31-s + 6·35-s + 2·37-s + 6·41-s + 4·43-s + 6·47-s − 3·49-s − 53-s − 6·55-s − 3·59-s + 13·67-s − 8·71-s + 9·73-s − 4·77-s − 79-s + 2·83-s + 6·85-s − 9·89-s − 15·95-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 0.755·7-s + 0.603·11-s − 0.485·17-s + 1.14·19-s + 1.66·23-s + 4/5·25-s − 1.48·29-s − 0.898·31-s + 1.01·35-s + 0.328·37-s + 0.937·41-s + 0.609·43-s + 0.875·47-s − 3/7·49-s − 0.137·53-s − 0.809·55-s − 0.390·59-s + 1.58·67-s − 0.949·71-s + 1.05·73-s − 0.455·77-s − 0.112·79-s + 0.219·83-s + 0.650·85-s − 0.953·89-s − 1.53·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 397 | \( 1 + T \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 - 9 T + p T^{2} \) | 1.97.aj |
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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
−13.00168527402078, −12.68036516789105, −12.33101375082372, −11.61271390824695, −11.38931997661751, −10.96407080111201, −10.59155711723364, −9.650989353289821, −9.401947315436000, −9.068450110885687, −8.497591413141851, −7.852181504740513, −7.400277772193589, −7.132524824998759, −6.643569453167317, −5.966951390617376, −5.459648087573782, −4.893375269905083, −4.260320116855018, −3.785575321400082, −3.417020245198942, −2.898623271941948, −2.207576495699467, −1.286901345366400, −0.7134653202421555, 0,
0.7134653202421555, 1.286901345366400, 2.207576495699467, 2.898623271941948, 3.417020245198942, 3.785575321400082, 4.260320116855018, 4.893375269905083, 5.459648087573782, 5.966951390617376, 6.643569453167317, 7.132524824998759, 7.400277772193589, 7.852181504740513, 8.497591413141851, 9.068450110885687, 9.401947315436000, 9.650989353289821, 10.59155711723364, 10.96407080111201, 11.38931997661751, 11.61271390824695, 12.33101375082372, 12.68036516789105, 13.00168527402078
