L(s) = 1 | + (1.36 − 1.36i)2-s + (−0.5 + 0.866i)3-s − 2.73i·4-s + (0.499 + 1.86i)6-s + (−2.36 − 2.36i)8-s + (−0.499 − 0.866i)9-s + (2.36 + 1.36i)12-s − i·13-s − 3.73·16-s + (−1.86 − 0.5i)18-s − i·23-s + (3.23 − 0.866i)24-s + 25-s + (−1.36 − 1.36i)26-s + 0.999·27-s + ⋯ |
L(s) = 1 | + (1.36 − 1.36i)2-s + (−0.5 + 0.866i)3-s − 2.73i·4-s + (0.499 + 1.86i)6-s + (−2.36 − 2.36i)8-s + (−0.499 − 0.866i)9-s + (2.36 + 1.36i)12-s − i·13-s − 3.73·16-s + (−1.86 − 0.5i)18-s − i·23-s + (3.23 − 0.866i)24-s + 25-s + (−1.36 − 1.36i)26-s + 0.999·27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.804499159\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.804499159\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 - 0.866i)T \) |
| 23 | \( 1 + iT \) |
| 29 | \( 1 + (0.866 + 0.5i)T \) |
good | 2 | \( 1 + (-1.36 + 1.36i)T - iT^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + iT^{2} \) |
| 13 | \( 1 + iT - T^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 31 | \( 1 + (0.366 + 0.366i)T + iT^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + (-0.366 - 0.366i)T + iT^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (-1.36 - 1.36i)T + iT^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - 2iT - T^{2} \) |
| 61 | \( 1 - iT^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + 1.73T + T^{2} \) |
| 73 | \( 1 + (0.366 - 0.366i)T - iT^{2} \) |
| 79 | \( 1 - iT^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + iT^{2} \) |
| 97 | \( 1 + iT^{2} \) |
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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
−9.435702103098115619321157900280, −8.718497816846362269557616129836, −7.18561845736582172009699467884, −5.90195641384237950891184622507, −5.75929918737616251336155296170, −4.65871248255268265131606166992, −4.20239051079156926265511367031, −3.20905929515802293840864512172, −2.52186702690588701174707878200, −0.906663652201849071416924115675,
2.02217805169350827756242471509, 3.24115687234898642842969228624, 4.20312680401528447391217630653, 5.14137690229698200961062988902, 5.66790195661538608611410497912, 6.50566644860060250791129497869, 7.14585695258219659786175950834, 7.53445030012669371307274076689, 8.555156390291384607004176566470, 9.146594178449210740008320538972