| L(s) = 1 | − 3-s + 3·7-s − 2·9-s + 2·13-s + 7·17-s − 5·19-s − 3·21-s + 2·23-s + 5·27-s − 3·29-s − 5·31-s + 5·37-s − 2·39-s − 2·41-s + 8·43-s + 10·47-s + 2·49-s − 7·51-s + 53-s + 5·57-s + 2·59-s + 7·61-s − 6·63-s − 8·67-s − 2·69-s + 71-s − 14·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.13·7-s − 2/3·9-s + 0.554·13-s + 1.69·17-s − 1.14·19-s − 0.654·21-s + 0.417·23-s + 0.962·27-s − 0.557·29-s − 0.898·31-s + 0.821·37-s − 0.320·39-s − 0.312·41-s + 1.21·43-s + 1.45·47-s + 2/7·49-s − 0.980·51-s + 0.137·53-s + 0.662·57-s + 0.260·59-s + 0.896·61-s − 0.755·63-s − 0.977·67-s − 0.240·69-s + 0.118·71-s − 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 - T + p T^{2} \) | 1.89.ab |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−13.23336672200209, −12.81669678527216, −12.31378483642466, −11.77786801555934, −11.51974780688557, −10.98964164703156, −10.54181621549750, −10.35741849184152, −9.454002650867425, −9.020431138959908, −8.540823930226065, −8.111943637100500, −7.540182949384574, −7.257513864562720, −6.386832374466915, −5.955424011955570, −5.530655861531309, −5.178390668970023, −4.498224127486923, −3.990723769302871, −3.418189922949889, −2.691341125088013, −2.139471583451775, −1.333740412097354, −0.9156315560299503, 0,
0.9156315560299503, 1.333740412097354, 2.139471583451775, 2.691341125088013, 3.418189922949889, 3.990723769302871, 4.498224127486923, 5.178390668970023, 5.530655861531309, 5.955424011955570, 6.386832374466915, 7.257513864562720, 7.540182949384574, 8.111943637100500, 8.540823930226065, 9.020431138959908, 9.454002650867425, 10.35741849184152, 10.54181621549750, 10.98964164703156, 11.51974780688557, 11.77786801555934, 12.31378483642466, 12.81669678527216, 13.23336672200209
