| L(s) = 1 | + 4·7-s − 4·11-s + 6·13-s − 8·17-s − 4·19-s + 8·23-s − 5·25-s − 2·29-s − 8·31-s − 10·37-s + 12·41-s + 4·43-s + 12·47-s + 9·49-s − 4·53-s + 6·59-s − 6·61-s + 4·67-s − 14·71-s − 6·73-s − 16·77-s + 8·79-s − 16·83-s + 12·89-s + 24·91-s − 18·97-s + 101-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 1.20·11-s + 1.66·13-s − 1.94·17-s − 0.917·19-s + 1.66·23-s − 25-s − 0.371·29-s − 1.43·31-s − 1.64·37-s + 1.87·41-s + 0.609·43-s + 1.75·47-s + 9/7·49-s − 0.549·53-s + 0.781·59-s − 0.768·61-s + 0.488·67-s − 1.66·71-s − 0.702·73-s − 1.82·77-s + 0.900·79-s − 1.75·83-s + 1.27·89-s + 2.51·91-s − 1.82·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.993438424\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.993438424\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 397 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 8 T + p T^{2} \) | 1.17.i |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−12.97852430943750, −12.64170389626456, −11.91367861811992, −11.32332664593477, −10.87017112848506, −10.74861306255261, −10.67401446718398, −9.466575847343789, −9.047193421695310, −8.618362804519469, −8.402096318429235, −7.743928468006261, −7.234837825691284, −6.926023265855704, −6.035177334464713, −5.731074043851126, −5.223645172367713, −4.621350500686631, −4.167557667663724, −3.790802027824036, −2.873037859908836, −2.320341996133293, −1.818193519464145, −1.290955658011308, −0.3832710303305036,
0.3832710303305036, 1.290955658011308, 1.818193519464145, 2.320341996133293, 2.873037859908836, 3.790802027824036, 4.167557667663724, 4.621350500686631, 5.223645172367713, 5.731074043851126, 6.035177334464713, 6.926023265855704, 7.234837825691284, 7.743928468006261, 8.402096318429235, 8.618362804519469, 9.047193421695310, 9.466575847343789, 10.67401446718398, 10.74861306255261, 10.87017112848506, 11.32332664593477, 11.91367861811992, 12.64170389626456, 12.97852430943750
