| L(s) = 1 | − 3-s + 2·5-s + 2·7-s + 9-s − 5·11-s − 3·13-s − 2·15-s − 3·17-s + 2·19-s − 2·21-s + 23-s − 25-s − 27-s + 5·31-s + 5·33-s + 4·35-s − 8·37-s + 3·39-s − 7·41-s − 43-s + 2·45-s + 8·47-s − 3·49-s + 3·51-s − 3·53-s − 10·55-s − 2·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 0.755·7-s + 1/3·9-s − 1.50·11-s − 0.832·13-s − 0.516·15-s − 0.727·17-s + 0.458·19-s − 0.436·21-s + 0.208·23-s − 1/5·25-s − 0.192·27-s + 0.898·31-s + 0.870·33-s + 0.676·35-s − 1.31·37-s + 0.480·39-s − 1.09·41-s − 0.152·43-s + 0.298·45-s + 1.16·47-s − 3/7·49-s + 0.420·51-s − 0.412·53-s − 1.34·55-s − 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.609513482\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.609513482\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 43 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 7 T + p T^{2} \) | 1.41.h |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 15 T + p T^{2} \) | 1.67.p |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 - 12 T + p T^{2} \) | 1.73.am |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 - 15 T + p T^{2} \) | 1.83.ap |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 11 T + p T^{2} \) | 1.97.al |
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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
−7.83620611179975402046355103066, −7.04321965881792285596276974177, −6.43639910106487022682301856073, −5.48930634298801492338176622242, −5.13526548711189134413027236604, −4.65548760803075054478115034619, −3.45351787730319045962736533071, −2.35178367710969705429993917568, −1.95421240095766874171702589267, −0.62223646975327610756695530290,
0.62223646975327610756695530290, 1.95421240095766874171702589267, 2.35178367710969705429993917568, 3.45351787730319045962736533071, 4.65548760803075054478115034619, 5.13526548711189134413027236604, 5.48930634298801492338176622242, 6.43639910106487022682301856073, 7.04321965881792285596276974177, 7.83620611179975402046355103066
