| L(s) = 1 | − 4·7-s − 3·9-s + 4·13-s − 8·19-s + 4·23-s − 5·25-s + 4·31-s + 8·37-s − 6·41-s − 8·43-s + 8·47-s + 9·49-s − 12·53-s − 8·59-s + 8·61-s + 12·63-s + 8·67-s + 12·71-s + 6·73-s − 12·79-s + 9·81-s + 8·83-s − 6·89-s − 16·91-s + 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 9-s + 1.10·13-s − 1.83·19-s + 0.834·23-s − 25-s + 0.718·31-s + 1.31·37-s − 0.937·41-s − 1.21·43-s + 1.16·47-s + 9/7·49-s − 1.64·53-s − 1.04·59-s + 1.02·61-s + 1.51·63-s + 0.977·67-s + 1.42·71-s + 0.702·73-s − 1.35·79-s + 81-s + 0.878·83-s − 0.635·89-s − 1.67·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−14.21937601856145, −13.76376742876888, −13.24568109445026, −12.93921468885766, −12.45481725728272, −11.81341971665362, −11.28727511643725, −10.81757035725035, −10.38752248512691, −9.650582212902614, −9.359241063366588, −8.718843021781006, −8.256198753288693, −7.887838830928272, −6.766679730026452, −6.582450715228590, −6.129521603057934, −5.636720001042705, −4.903205826436101, −4.075822118589312, −3.675119003570595, −3.012811971774782, −2.550496002369746, −1.741688703743571, −0.6970964892291323, 0,
0.6970964892291323, 1.741688703743571, 2.550496002369746, 3.012811971774782, 3.675119003570595, 4.075822118589312, 4.903205826436101, 5.636720001042705, 6.129521603057934, 6.582450715228590, 6.766679730026452, 7.887838830928272, 8.256198753288693, 8.718843021781006, 9.359241063366588, 9.650582212902614, 10.38752248512691, 10.81757035725035, 11.28727511643725, 11.81341971665362, 12.45481725728272, 12.93921468885766, 13.24568109445026, 13.76376742876888, 14.21937601856145
