| L(s) = 1 | − 3-s − 4·5-s − 2·9-s − 13-s + 4·15-s + 3·17-s + 6·19-s − 7·23-s + 11·25-s + 5·27-s − 9·29-s + 8·31-s − 2·37-s + 39-s − 8·41-s − 7·43-s + 8·45-s − 2·47-s − 7·49-s − 3·51-s + 9·53-s − 6·57-s + 4·59-s − 7·61-s + 4·65-s + 8·67-s + 7·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.78·5-s − 2/3·9-s − 0.277·13-s + 1.03·15-s + 0.727·17-s + 1.37·19-s − 1.45·23-s + 11/5·25-s + 0.962·27-s − 1.67·29-s + 1.43·31-s − 0.328·37-s + 0.160·39-s − 1.24·41-s − 1.06·43-s + 1.19·45-s − 0.291·47-s − 49-s − 0.420·51-s + 1.23·53-s − 0.794·57-s + 0.520·59-s − 0.896·61-s + 0.496·65-s + 0.977·67-s + 0.842·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 7 T + p T^{2} \) | 1.23.h |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 5 T + p T^{2} \) | 1.79.af |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.82031702038198, −13.76405010985283, −12.77712658629369, −12.27022298280617, −11.94019764416967, −11.62107642605548, −11.29153027263311, −10.72528887103307, −10.03353902753111, −9.697377914979525, −8.950155136632278, −8.265770069013650, −7.925321488357276, −7.736576668786755, −6.760843192012378, −6.684062340919366, −5.708310919918773, −5.199867446301315, −4.911690870295640, −3.989757706582788, −3.624706083632888, −3.173168813345393, −2.415588051119683, −1.410113892113636, −0.5987918597381158, 0,
0.5987918597381158, 1.410113892113636, 2.415588051119683, 3.173168813345393, 3.624706083632888, 3.989757706582788, 4.911690870295640, 5.199867446301315, 5.708310919918773, 6.684062340919366, 6.760843192012378, 7.736576668786755, 7.925321488357276, 8.265770069013650, 8.950155136632278, 9.697377914979525, 10.03353902753111, 10.72528887103307, 11.29153027263311, 11.62107642605548, 11.94019764416967, 12.27022298280617, 12.77712658629369, 13.76405010985283, 13.82031702038198
