L(s) = 1 | + (0.707 − 0.707i)2-s − 1.00i·4-s + 7-s + (−0.707 − 0.707i)8-s + (0.707 − 0.707i)14-s − 1.00·16-s + 1.41i·23-s − i·25-s − 1.00i·28-s + (−1.41 − 1.41i)29-s + (−0.707 + 0.707i)32-s + (1 + i)37-s + (−1 + i)43-s + (1.00 + 1.00i)46-s + 49-s + (−0.707 − 0.707i)50-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)2-s − 1.00i·4-s + 7-s + (−0.707 − 0.707i)8-s + (0.707 − 0.707i)14-s − 1.00·16-s + 1.41i·23-s − i·25-s − 1.00i·28-s + (−1.41 − 1.41i)29-s + (−0.707 + 0.707i)32-s + (1 + i)37-s + (−1 + i)43-s + (1.00 + 1.00i)46-s + 49-s + (−0.707 − 0.707i)50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.220 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.220 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.537674373\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.537674373\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 + iT^{2} \) |
| 11 | \( 1 + iT^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 23 | \( 1 - 1.41iT - T^{2} \) |
| 29 | \( 1 + (1.41 + 1.41i)T + iT^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (-1 - i)T + iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + (1 - i)T - iT^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 - iT^{2} \) |
| 61 | \( 1 + iT^{2} \) |
| 67 | \( 1 + (1 + i)T + iT^{2} \) |
| 71 | \( 1 - 1.41iT - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - 2iT - T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.03558612072646599533483768018, −9.472624149844286240490062680954, −8.329411204426083140383931178926, −7.51657683477886130113432645363, −6.31483394464622133943857267186, −5.49422473080166745309321938067, −4.62259854587360629243389405516, −3.79550733665516475839717743104, −2.53658919254501107175928967647, −1.44738217140645465853481811861,
1.99423349873421468188947493563, 3.33251079663840649548972603794, 4.37609257043380744193248954499, 5.16220407483775448892857358292, 5.94633293141694816573895335594, 7.06651803605067525795674264846, 7.63158143063594481271876513483, 8.586913285695127649417628656423, 9.157356584867682468761528696308, 10.56904680234079532896452384842