| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 13-s − 14-s + 16-s + 3·17-s + 19-s + 6·23-s − 26-s − 28-s + 3·29-s − 4·31-s + 32-s + 3·34-s + 37-s + 38-s − 4·43-s + 6·46-s + 6·47-s − 6·49-s − 52-s + 6·53-s − 56-s + 3·58-s + 6·59-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s + 0.229·19-s + 1.25·23-s − 0.196·26-s − 0.188·28-s + 0.557·29-s − 0.718·31-s + 0.176·32-s + 0.514·34-s + 0.164·37-s + 0.162·38-s − 0.609·43-s + 0.884·46-s + 0.875·47-s − 6/7·49-s − 0.138·52-s + 0.824·53-s − 0.133·56-s + 0.393·58-s + 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.033792786\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.033792786\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−14.41408110557122, −13.92321322174746, −13.40546368384149, −12.79120661815596, −12.59362762727967, −11.94243593595047, −11.40137681972468, −11.00544856606620, −10.26409341830456, −9.919689303526687, −9.294978965115833, −8.695966778517229, −8.089984630514672, −7.444985636550450, −6.954000204382730, −6.513304848101884, −5.775695729213451, −5.237433791257774, −4.871509534289127, −3.978921423076606, −3.542459771625302, −2.866862193874471, −2.328961406333248, −1.392243143506751, −0.6395027469292625,
0.6395027469292625, 1.392243143506751, 2.328961406333248, 2.866862193874471, 3.542459771625302, 3.978921423076606, 4.871509534289127, 5.237433791257774, 5.775695729213451, 6.513304848101884, 6.954000204382730, 7.444985636550450, 8.089984630514672, 8.695966778517229, 9.294978965115833, 9.919689303526687, 10.26409341830456, 11.00544856606620, 11.40137681972468, 11.94243593595047, 12.59362762727967, 12.79120661815596, 13.40546368384149, 13.92321322174746, 14.41408110557122
