L(s) = 1 | + (1.36 − 0.366i)2-s + (0.866 − 0.5i)4-s + (1 + i)5-s + (1.73 + i)10-s + (−0.366 − 1.36i)11-s + (−0.499 + 0.866i)16-s + (1.36 + 0.366i)20-s + (−1 − 1.73i)22-s + i·25-s + (−0.366 + 1.36i)32-s + (−1.36 + 0.366i)41-s + (1.73 − i)43-s + (−1 − 0.999i)44-s + (−1 + i)47-s + (−0.866 − 0.5i)49-s + (0.366 + 1.36i)50-s + ⋯ |
L(s) = 1 | + (1.36 − 0.366i)2-s + (0.866 − 0.5i)4-s + (1 + i)5-s + (1.73 + i)10-s + (−0.366 − 1.36i)11-s + (−0.499 + 0.866i)16-s + (1.36 + 0.366i)20-s + (−1 − 1.73i)22-s + i·25-s + (−0.366 + 1.36i)32-s + (−1.36 + 0.366i)41-s + (1.73 − i)43-s + (−1 − 0.999i)44-s + (−1 + i)47-s + (−0.866 − 0.5i)49-s + (0.366 + 1.36i)50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0386i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0386i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.427140141\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.427140141\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 5 | \( 1 + (-1 - i)T + iT^{2} \) |
| 7 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + iT^{2} \) |
| 37 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 43 | \( 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (1 - i)T - iT^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + (1 + i)T + iT^{2} \) |
| 89 | \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 97 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
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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
−9.880872982848770537550837972148, −8.937656671001005100145853633412, −8.009555252667046027492794295819, −6.79381963602542819335994620996, −6.10024487046170048702413973472, −5.61110077237492478153073272798, −4.66548507964020217502891770608, −3.41862342013857292150232990428, −2.94371747653891445625218477173, −1.93216772732673882481633639646,
1.66788522944546560811375725694, 2.75928812159246748507520268650, 4.07800100687861798798968593928, 4.84672972981776856958144039799, 5.31946055384630044631537323867, 6.16528671611276756694382475602, 6.95226163330634574248551785687, 7.87894053040409549726745463023, 9.010093944693941763344673284335, 9.627295815795258291399525624070