| L(s) = 1 | + 2-s + 4-s + 2·5-s + 8-s + 2·10-s − 11-s + 13-s + 16-s − 2·17-s + 2·20-s − 22-s + 4·23-s − 25-s + 26-s − 2·29-s + 32-s − 2·34-s − 2·37-s + 2·40-s + 6·41-s + 8·43-s − 44-s + 4·46-s − 12·47-s − 50-s + 52-s − 2·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.353·8-s + 0.632·10-s − 0.301·11-s + 0.277·13-s + 1/4·16-s − 0.485·17-s + 0.447·20-s − 0.213·22-s + 0.834·23-s − 1/5·25-s + 0.196·26-s − 0.371·29-s + 0.176·32-s − 0.342·34-s − 0.328·37-s + 0.316·40-s + 0.937·41-s + 1.21·43-s − 0.150·44-s + 0.589·46-s − 1.75·47-s − 0.141·50-s + 0.138·52-s − 0.274·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 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.c |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.59031432265879, −13.25324495918780, −13.00956732565786, −12.48257661281627, −11.81676983879774, −11.39477257267909, −10.89042727975455, −10.44908893364755, −9.961874864813859, −9.339971252687751, −9.011470084689479, −8.351213278419578, −7.758454864659506, −7.196711639557166, −6.742558094941666, −6.090879146376830, −5.750592974788999, −5.308653714160732, −4.531959209173892, −4.294094142728290, −3.371250731529626, −2.948519856965470, −2.283967742806143, −1.734149272215953, −1.084239440057077, 0,
1.084239440057077, 1.734149272215953, 2.283967742806143, 2.948519856965470, 3.371250731529626, 4.294094142728290, 4.531959209173892, 5.308653714160732, 5.750592974788999, 6.090879146376830, 6.742558094941666, 7.196711639557166, 7.758454864659506, 8.351213278419578, 9.011470084689479, 9.339971252687751, 9.961874864813859, 10.44908893364755, 10.89042727975455, 11.39477257267909, 11.81676983879774, 12.48257661281627, 13.00956732565786, 13.25324495918780, 13.59031432265879
