| L(s) = 1 | − 3-s − 2·7-s + 9-s + 4·11-s − 13-s − 6·17-s + 5·19-s + 2·21-s + 6·23-s − 27-s + 29-s − 10·31-s − 4·33-s − 3·37-s + 39-s − 7·41-s + 8·43-s + 7·47-s − 3·49-s + 6·51-s + 3·53-s − 5·57-s − 6·59-s − 2·61-s − 2·63-s + 9·67-s − 6·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s + 1.20·11-s − 0.277·13-s − 1.45·17-s + 1.14·19-s + 0.436·21-s + 1.25·23-s − 0.192·27-s + 0.185·29-s − 1.79·31-s − 0.696·33-s − 0.493·37-s + 0.160·39-s − 1.09·41-s + 1.21·43-s + 1.02·47-s − 3/7·49-s + 0.840·51-s + 0.412·53-s − 0.662·57-s − 0.781·59-s − 0.256·61-s − 0.251·63-s + 1.09·67-s − 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + 7 T + p T^{2} \) | 1.41.h |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 9 T + p T^{2} \) | 1.67.aj |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−13.70026432725421, −13.24150549475019, −12.73180205150372, −12.38009690128753, −11.70810866497393, −11.50571592418934, −10.77773980605227, −10.60597654820285, −9.770880783473194, −9.305769287116779, −9.064176433356755, −8.597632081389222, −7.643460511562078, −7.119764343241377, −6.879694407355892, −6.360054637868383, −5.742840361612654, −5.283103190244607, −4.622940382168507, −4.128150521413456, −3.473588594995206, −3.026781733075350, −2.165632825869307, −1.509398301699228, −0.7806578742040454, 0,
0.7806578742040454, 1.509398301699228, 2.165632825869307, 3.026781733075350, 3.473588594995206, 4.128150521413456, 4.622940382168507, 5.283103190244607, 5.742840361612654, 6.360054637868383, 6.879694407355892, 7.119764343241377, 7.643460511562078, 8.597632081389222, 9.064176433356755, 9.305769287116779, 9.770880783473194, 10.60597654820285, 10.77773980605227, 11.50571592418934, 11.70810866497393, 12.38009690128753, 12.73180205150372, 13.24150549475019, 13.70026432725421
