| L(s) = 1 | + 2-s + 4-s + 8-s + 3·11-s + 2·13-s + 16-s + 3·17-s + 19-s + 3·22-s − 6·23-s + 2·26-s − 6·29-s + 4·31-s + 32-s + 3·34-s + 4·37-s + 38-s + 9·41-s + 43-s + 3·44-s − 6·46-s + 6·47-s + 2·52-s + 12·53-s − 6·58-s + 3·59-s − 8·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 0.904·11-s + 0.554·13-s + 1/4·16-s + 0.727·17-s + 0.229·19-s + 0.639·22-s − 1.25·23-s + 0.392·26-s − 1.11·29-s + 0.718·31-s + 0.176·32-s + 0.514·34-s + 0.657·37-s + 0.162·38-s + 1.40·41-s + 0.152·43-s + 0.452·44-s − 0.884·46-s + 0.875·47-s + 0.277·52-s + 1.64·53-s − 0.787·58-s + 0.390·59-s − 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.916188174\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.916188174\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 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.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−13.13289349908185, −12.55953371326226, −12.07958948195344, −11.86404088116265, −11.20711193990728, −10.94545570580812, −10.22793862506865, −9.865525270421623, −9.283667734071323, −8.878563955971528, −8.195405888603090, −7.738687651627462, −7.316001898429731, −6.716016431115546, −6.105468165628449, −5.857943546780580, −5.373736591994287, −4.623034094192918, −3.980162968102356, −3.883207376326452, −3.169327662930728, −2.481377454843634, −1.943510904591519, −1.196398563733638, −0.6526686050233431,
0.6526686050233431, 1.196398563733638, 1.943510904591519, 2.481377454843634, 3.169327662930728, 3.883207376326452, 3.980162968102356, 4.623034094192918, 5.373736591994287, 5.857943546780580, 6.105468165628449, 6.716016431115546, 7.316001898429731, 7.738687651627462, 8.195405888603090, 8.878563955971528, 9.283667734071323, 9.865525270421623, 10.22793862506865, 10.94545570580812, 11.20711193990728, 11.86404088116265, 12.07958948195344, 12.55953371326226, 13.13289349908185
