L(s) = 1 | + (0.866 − 0.5i)2-s + (−0.707 + 0.707i)3-s + (0.499 − 0.866i)4-s + (−0.965 + 0.258i)5-s + (−0.258 + 0.965i)6-s + (0.258 − 0.965i)7-s − 0.999i·8-s − 1.00i·9-s + (−0.707 + 0.707i)10-s + (−0.866 + 0.5i)11-s + (0.258 + 0.965i)12-s + (0.965 − 0.258i)13-s + (−0.258 − 0.965i)14-s + (0.500 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (0.707 − 0.707i)17-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)2-s + (−0.707 + 0.707i)3-s + (0.499 − 0.866i)4-s + (−0.965 + 0.258i)5-s + (−0.258 + 0.965i)6-s + (0.258 − 0.965i)7-s − 0.999i·8-s − 1.00i·9-s + (−0.707 + 0.707i)10-s + (−0.866 + 0.5i)11-s + (0.258 + 0.965i)12-s + (0.965 − 0.258i)13-s + (−0.258 − 0.965i)14-s + (0.500 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (0.707 − 0.707i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.203 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.203 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.149587184\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.149587184\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (0.965 - 0.258i)T \) |
| 7 | \( 1 + (-0.258 + 0.965i)T \) |
good | 11 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 19 | \( 1 + 1.41iT - T^{2} \) |
| 23 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \) |
| 47 | \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 + (-1 + i)T - iT^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 + iT - T^{2} \) |
| 73 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \) |
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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
−10.17168426143056433441179524449, −9.190230491339370097188236890127, −7.84639230471449711013703873253, −7.08396354479650604797513374983, −6.26496274422130634882529890788, −5.05027371378681590053794638238, −4.62484155320102833649718552670, −3.69712535294971603019438698963, −2.93017118210419894478014572689, −0.867606496431235361434184519123,
1.78055617522271938759178022443, 3.17533921904129546712277527392, 4.14187845862578180575169938051, 5.35540660959799711955312572719, 5.69860792824048243877858219393, 6.58510323023082012269725296861, 7.65619554752602771627344333669, 8.173818691072038166200057846396, 8.686782360734667004615560529366, 10.47954814372974577938475977838