| L(s) = 1 | + 3-s − 3·7-s + 9-s + 11-s − 4·13-s − 5·17-s + 7·19-s − 3·21-s − 9·23-s + 27-s − 29-s + 2·31-s + 33-s + 5·37-s − 4·39-s − 9·41-s + 4·43-s − 7·47-s + 2·49-s − 5·51-s + 7·57-s + 3·59-s + 8·61-s − 3·63-s + 4·67-s − 9·69-s − 3·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.13·7-s + 1/3·9-s + 0.301·11-s − 1.10·13-s − 1.21·17-s + 1.60·19-s − 0.654·21-s − 1.87·23-s + 0.192·27-s − 0.185·29-s + 0.359·31-s + 0.174·33-s + 0.821·37-s − 0.640·39-s − 1.40·41-s + 0.609·43-s − 1.02·47-s + 2/7·49-s − 0.700·51-s + 0.927·57-s + 0.390·59-s + 1.02·61-s − 0.377·63-s + 0.488·67-s − 1.08·69-s − 0.356·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 382800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 382800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6604962594\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6604962594\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 29 | \( 1 + T \) | |
| good | 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 + 5 T + p T^{2} \) | 1.79.f |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 97 | \( 1 + 3 T + p T^{2} \) | 1.97.d |
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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
−12.66677644913461, −11.88063634834494, −11.71741771961596, −11.30332328552359, −10.39305356382410, −10.04557653059485, −9.814755611022082, −9.350246286099145, −8.963239090114015, −8.399924515868141, −7.789794190857590, −7.547676528470712, −6.922076156047523, −6.477964641504654, −6.190099449596591, −5.420832738664111, −4.983925340174768, −4.409270955103871, −3.739775526537050, −3.565558720072277, −2.719402686301974, −2.493445807075289, −1.839812530508521, −1.116767610891206, −0.2028141590042152,
0.2028141590042152, 1.116767610891206, 1.839812530508521, 2.493445807075289, 2.719402686301974, 3.565558720072277, 3.739775526537050, 4.409270955103871, 4.983925340174768, 5.420832738664111, 6.190099449596591, 6.477964641504654, 6.922076156047523, 7.547676528470712, 7.789794190857590, 8.399924515868141, 8.963239090114015, 9.350246286099145, 9.814755611022082, 10.04557653059485, 10.39305356382410, 11.30332328552359, 11.71741771961596, 11.88063634834494, 12.66677644913461
