| L(s) = 1 | + 5-s − 4·7-s + 11-s − 4·13-s + 6·17-s + 2·19-s + 25-s − 4·31-s − 4·35-s − 10·37-s − 4·43-s − 12·47-s + 9·49-s − 6·53-s + 55-s − 12·59-s − 10·61-s − 4·65-s − 4·67-s + 8·73-s − 4·77-s − 10·79-s + 6·83-s + 6·85-s + 6·89-s + 16·91-s + 2·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.51·7-s + 0.301·11-s − 1.10·13-s + 1.45·17-s + 0.458·19-s + 1/5·25-s − 0.718·31-s − 0.676·35-s − 1.64·37-s − 0.609·43-s − 1.75·47-s + 9/7·49-s − 0.824·53-s + 0.134·55-s − 1.56·59-s − 1.28·61-s − 0.496·65-s − 0.488·67-s + 0.936·73-s − 0.455·77-s − 1.12·79-s + 0.658·83-s + 0.650·85-s + 0.635·89-s + 1.67·91-s + 0.205·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−9.038549791749297259415708863685, −7.896053225312047578470722141565, −7.12289288365075576894664921496, −6.43361771992124639760583933441, −5.62904260526368110524318216143, −4.84063087230686996605693150347, −3.46288274003084083808791132864, −3.01690060177596489931621497136, −1.63354959955777809280108165155, 0,
1.63354959955777809280108165155, 3.01690060177596489931621497136, 3.46288274003084083808791132864, 4.84063087230686996605693150347, 5.62904260526368110524318216143, 6.43361771992124639760583933441, 7.12289288365075576894664921496, 7.896053225312047578470722141565, 9.038549791749297259415708863685
