| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s − 13-s + 15-s − 4·17-s + 3·19-s + 21-s − 6·23-s − 4·25-s − 27-s − 7·29-s − 4·31-s + 35-s + 37-s + 39-s + 4·41-s − 2·43-s − 45-s − 7·47-s + 49-s + 4·51-s + 10·53-s − 3·57-s + 9·59-s + 2·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.277·13-s + 0.258·15-s − 0.970·17-s + 0.688·19-s + 0.218·21-s − 1.25·23-s − 4/5·25-s − 0.192·27-s − 1.29·29-s − 0.718·31-s + 0.169·35-s + 0.164·37-s + 0.160·39-s + 0.624·41-s − 0.304·43-s − 0.149·45-s − 1.02·47-s + 1/7·49-s + 0.560·51-s + 1.37·53-s − 0.397·57-s + 1.17·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 7 T + p T^{2} \) | 1.29.h |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 9 T + p T^{2} \) | 1.67.aj |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 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
−15.12628408383029, −14.61124636691851, −13.84338211048202, −13.49071884459585, −12.85690379623675, −12.41229722313769, −11.86779975132808, −11.25547089300016, −11.13318820609561, −10.25545598527374, −9.727460030391878, −9.407427062262207, −8.615675207667735, −8.005283640186272, −7.473477698580643, −6.944694173345396, −6.365532557269228, −5.717283937927409, −5.267332659095005, −4.520087239838018, −3.788039492329146, −3.558272437680499, −2.334206718846888, −1.934427477885446, −0.7341433067831738, 0,
0.7341433067831738, 1.934427477885446, 2.334206718846888, 3.558272437680499, 3.788039492329146, 4.520087239838018, 5.267332659095005, 5.717283937927409, 6.365532557269228, 6.944694173345396, 7.473477698580643, 8.005283640186272, 8.615675207667735, 9.407427062262207, 9.727460030391878, 10.25545598527374, 11.13318820609561, 11.25547089300016, 11.86779975132808, 12.41229722313769, 12.85690379623675, 13.49071884459585, 13.84338211048202, 14.61124636691851, 15.12628408383029
