| L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 10-s + 13-s + 16-s + 6·17-s + 20-s + 4·23-s + 25-s − 26-s + 2·29-s − 32-s − 6·34-s + 6·37-s − 40-s − 10·41-s − 8·43-s − 4·46-s − 12·47-s − 7·49-s − 50-s + 52-s + 6·53-s − 2·58-s + 12·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s + 0.277·13-s + 1/4·16-s + 1.45·17-s + 0.223·20-s + 0.834·23-s + 1/5·25-s − 0.196·26-s + 0.371·29-s − 0.176·32-s − 1.02·34-s + 0.986·37-s − 0.158·40-s − 1.56·41-s − 1.21·43-s − 0.589·46-s − 1.75·47-s − 49-s − 0.141·50-s + 0.138·52-s + 0.824·53-s − 0.262·58-s + 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.55708731620243, −13.01387261240090, −12.87545118755331, −11.98975324955165, −11.65753573661636, −11.33103352734270, −10.57717774043225, −10.14288479040236, −9.850430393068596, −9.416391818392647, −8.670247456265449, −8.391993749715427, −7.885889021566143, −7.320504419121962, −6.673250165186001, −6.451351964549392, −5.689340090532816, −5.203011737076498, −4.802908026016330, −3.834907328242171, −3.314201174832180, −2.848047508915217, −2.085819611556633, −1.397286500851507, −0.9718695006837012, 0,
0.9718695006837012, 1.397286500851507, 2.085819611556633, 2.848047508915217, 3.314201174832180, 3.834907328242171, 4.802908026016330, 5.203011737076498, 5.689340090532816, 6.451351964549392, 6.673250165186001, 7.320504419121962, 7.885889021566143, 8.391993749715427, 8.670247456265449, 9.416391818392647, 9.850430393068596, 10.14288479040236, 10.57717774043225, 11.33103352734270, 11.65753573661636, 11.98975324955165, 12.87545118755331, 13.01387261240090, 13.55708731620243
