| L(s) = 1 | + 4·7-s + 11-s + 4·13-s − 6·17-s + 2·19-s − 4·31-s + 10·37-s + 4·43-s + 12·47-s + 9·49-s + 6·53-s − 12·59-s − 10·61-s + 4·67-s − 8·73-s + 4·77-s − 10·79-s − 6·83-s + 6·89-s + 16·91-s + 10·97-s + 12·101-s + 16·103-s + 18·107-s + 14·109-s − 6·113-s − 24·119-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 0.301·11-s + 1.10·13-s − 1.45·17-s + 0.458·19-s − 0.718·31-s + 1.64·37-s + 0.609·43-s + 1.75·47-s + 9/7·49-s + 0.824·53-s − 1.56·59-s − 1.28·61-s + 0.488·67-s − 0.936·73-s + 0.455·77-s − 1.12·79-s − 0.658·83-s + 0.635·89-s + 1.67·91-s + 1.01·97-s + 1.19·101-s + 1.57·103-s + 1.74·107-s + 1.34·109-s − 0.564·113-s − 2.20·119-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
| \(L(1)\) |
\(\approx\) |
\(2.859295638\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.859295638\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 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.ak |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 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.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−7.59196520383050307088764456269, −7.18895779369100601835410432634, −6.08739090114993401873713499957, −5.79873339296443103552506706726, −4.63750347994035412206500539666, −4.42928745039523348994919550629, −3.51585403649731787806164371474, −2.43376824322462399942956818820, −1.69530465422291409652423476652, −0.850346386070725796232646431876,
0.850346386070725796232646431876, 1.69530465422291409652423476652, 2.43376824322462399942956818820, 3.51585403649731787806164371474, 4.42928745039523348994919550629, 4.63750347994035412206500539666, 5.79873339296443103552506706726, 6.08739090114993401873713499957, 7.18895779369100601835410432634, 7.59196520383050307088764456269
