| L(s) = 1 | − 2·7-s + 11-s + 6·13-s − 2·17-s − 4·19-s + 4·23-s + 6·29-s − 8·31-s − 8·37-s + 2·41-s − 10·43-s − 12·47-s − 3·49-s + 8·53-s + 2·61-s + 4·67-s + 4·71-s + 10·73-s − 2·77-s − 16·79-s − 6·83-s − 2·89-s − 12·91-s + 4·97-s − 6·101-s + 8·103-s + 18·107-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 0.301·11-s + 1.66·13-s − 0.485·17-s − 0.917·19-s + 0.834·23-s + 1.11·29-s − 1.43·31-s − 1.31·37-s + 0.312·41-s − 1.52·43-s − 1.75·47-s − 3/7·49-s + 1.09·53-s + 0.256·61-s + 0.488·67-s + 0.474·71-s + 1.17·73-s − 0.227·77-s − 1.80·79-s − 0.658·83-s − 0.211·89-s − 1.25·91-s + 0.406·97-s − 0.597·101-s + 0.788·103-s + 1.74·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 8 T + p T^{2} \) | 1.53.ai |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
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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
−7.05488925862793760515073267802, −6.64556779530005623119082788665, −6.13663047200579074564891976846, −5.30874595906011267580010104753, −4.50954470626171265724338798069, −3.59953337893438484760664278550, −3.28105945033907310456619715307, −2.10157647242675039894071686789, −1.23555998271693626765307611825, 0,
1.23555998271693626765307611825, 2.10157647242675039894071686789, 3.28105945033907310456619715307, 3.59953337893438484760664278550, 4.50954470626171265724338798069, 5.30874595906011267580010104753, 6.13663047200579074564891976846, 6.64556779530005623119082788665, 7.05488925862793760515073267802
