| L(s) = 1 | + 3-s + 5·7-s + 9-s + 11-s + 4·13-s + 5·17-s + 7·19-s + 5·21-s + 9·23-s + 27-s − 2·29-s − 4·31-s + 33-s − 7·37-s + 4·39-s − 7·41-s − 9·47-s + 18·49-s + 5·51-s + 2·53-s + 7·57-s − 7·59-s − 2·61-s + 5·63-s − 2·67-s + 9·69-s − 5·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.88·7-s + 1/3·9-s + 0.301·11-s + 1.10·13-s + 1.21·17-s + 1.60·19-s + 1.09·21-s + 1.87·23-s + 0.192·27-s − 0.371·29-s − 0.718·31-s + 0.174·33-s − 1.15·37-s + 0.640·39-s − 1.09·41-s − 1.31·47-s + 18/7·49-s + 0.700·51-s + 0.274·53-s + 0.927·57-s − 0.911·59-s − 0.256·61-s + 0.629·63-s − 0.244·67-s + 1.08·69-s − 0.593·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.980255551\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.980255551\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 7 | \( 1 - 5 T + p T^{2} \) | 1.7.af |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + 7 T + p T^{2} \) | 1.41.h |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 7 T + p T^{2} \) | 1.59.h |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 3 T + p T^{2} \) | 1.79.ad |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 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
−14.50685264974616, −13.90599692861572, −13.66613264606865, −13.09352363696306, −12.29503113934342, −11.84680573646005, −11.39827760077167, −10.86901821465192, −10.51322068827927, −9.667926508441001, −9.186710178427495, −8.627268598627359, −8.255484212390261, −7.599919757808605, −7.330639008805286, −6.634597252075718, −5.708439108571887, −5.145858402927666, −4.953537297333649, −4.024564545963085, −3.363582478626697, −3.032917726245742, −1.829676777187721, −1.437484995369717, −0.9367103889308592,
0.9367103889308592, 1.437484995369717, 1.829676777187721, 3.032917726245742, 3.363582478626697, 4.024564545963085, 4.953537297333649, 5.145858402927666, 5.708439108571887, 6.634597252075718, 7.330639008805286, 7.599919757808605, 8.255484212390261, 8.627268598627359, 9.186710178427495, 9.667926508441001, 10.51322068827927, 10.86901821465192, 11.39827760077167, 11.84680573646005, 12.29503113934342, 13.09352363696306, 13.66613264606865, 13.90599692861572, 14.50685264974616
