| L(s) = 1 | − 2·4-s + 3·11-s − 4·13-s + 4·16-s − 7·17-s + 3·19-s + 5·23-s − 29-s + 31-s + 8·41-s − 10·43-s − 6·44-s − 2·47-s − 7·49-s + 8·52-s + 9·53-s + 12·59-s − 2·61-s − 8·64-s − 11·67-s + 14·68-s + 2·71-s + 10·73-s − 6·76-s − 10·79-s − 3·83-s − 7·89-s + ⋯ |
| L(s) = 1 | − 4-s + 0.904·11-s − 1.10·13-s + 16-s − 1.69·17-s + 0.688·19-s + 1.04·23-s − 0.185·29-s + 0.179·31-s + 1.24·41-s − 1.52·43-s − 0.904·44-s − 0.291·47-s − 49-s + 1.10·52-s + 1.23·53-s + 1.56·59-s − 0.256·61-s − 64-s − 1.34·67-s + 1.69·68-s + 0.237·71-s + 1.17·73-s − 0.688·76-s − 1.12·79-s − 0.329·83-s − 0.741·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 31 | \( 1 - T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 11 T + p T^{2} \) | 1.67.l |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 + 7 T + p T^{2} \) | 1.89.h |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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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
−7.53195204702960205591793355015, −6.97100420332807099229507858774, −6.23543165220367646793180589230, −5.26809099658662807591767952172, −4.73108283071087829693705035723, −4.11643549399312769040049033363, −3.26193995005782637557861464154, −2.29620745536765727397260527578, −1.13748345165350200951233232485, 0,
1.13748345165350200951233232485, 2.29620745536765727397260527578, 3.26193995005782637557861464154, 4.11643549399312769040049033363, 4.73108283071087829693705035723, 5.26809099658662807591767952172, 6.23543165220367646793180589230, 6.97100420332807099229507858774, 7.53195204702960205591793355015
