L(s) = 1 | + (0.766 − 0.642i)4-s + (−0.5 − 0.866i)7-s + (1.70 − 0.300i)13-s + (0.173 − 0.984i)16-s + (−0.173 − 0.984i)25-s + (−0.939 − 0.342i)28-s + (−1.5 + 0.866i)31-s + 1.73i·37-s + (−0.766 − 0.642i)43-s + (1.11 − 1.32i)52-s + (0.766 − 0.642i)61-s + (−0.500 − 0.866i)64-s + (0.592 − 1.62i)67-s + (−0.173 + 0.984i)73-s + (1.70 + 0.300i)79-s + ⋯ |
L(s) = 1 | + (0.766 − 0.642i)4-s + (−0.5 − 0.866i)7-s + (1.70 − 0.300i)13-s + (0.173 − 0.984i)16-s + (−0.173 − 0.984i)25-s + (−0.939 − 0.342i)28-s + (−1.5 + 0.866i)31-s + 1.73i·37-s + (−0.766 − 0.642i)43-s + (1.11 − 1.32i)52-s + (0.766 − 0.642i)61-s + (−0.500 − 0.866i)64-s + (0.592 − 1.62i)67-s + (−0.173 + 0.984i)73-s + (1.70 + 0.300i)79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.304 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.304 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.489551519\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.489551519\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 5 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-1.70 + 0.300i)T + (0.939 - 0.342i)T^{2} \) |
| 17 | \( 1 + (0.766 - 0.642i)T^{2} \) |
| 23 | \( 1 + (0.173 - 0.984i)T^{2} \) |
| 29 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 31 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - 1.73iT - T^{2} \) |
| 41 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 43 | \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \) |
| 47 | \( 1 + (0.766 + 0.642i)T^{2} \) |
| 53 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 59 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 61 | \( 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2} \) |
| 67 | \( 1 + (-0.592 + 1.62i)T + (-0.766 - 0.642i)T^{2} \) |
| 71 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 73 | \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \) |
| 79 | \( 1 + (-1.70 - 0.300i)T + (0.939 + 0.342i)T^{2} \) |
| 83 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 97 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
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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
−8.596160179251757025892925941648, −7.947444107414515212066435895506, −6.90501796400143512990243668982, −6.56906821600786214942760333741, −5.79958839474875599971165106542, −4.96002134659983117743639536372, −3.78601309154828107251850172803, −3.21427440428828665920101455504, −1.91832411437256368112242723041, −0.932078019798170067052399531276,
1.61075690811809028187190164740, 2.50788933762156103365399524622, 3.51161516678686829706705452382, 3.96993315372447799622206024359, 5.49102795903079912525543414623, 5.98007554947056407559081341054, 6.72187047388390880609152760521, 7.47830657588383029561815223714, 8.254962194027397371482410799334, 8.979122819354706931894465151542