L(s) = 1 | + (1 − i)2-s + i·3-s − i·4-s + (1 + i)6-s − 9-s + 12-s + 2i·13-s + 16-s + (−1 + i)18-s + i·23-s + 25-s + (2 + 2i)26-s − i·27-s − i·29-s + (−1 − i)31-s + (1 − i)32-s + ⋯ |
L(s) = 1 | + (1 − i)2-s + i·3-s − i·4-s + (1 + i)6-s − 9-s + 12-s + 2i·13-s + 16-s + (−1 + i)18-s + i·23-s + 25-s + (2 + 2i)26-s − i·27-s − i·29-s + (−1 − i)31-s + (1 − i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.933768823\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.933768823\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 23 | \( 1 - iT \) |
| 29 | \( 1 + iT \) |
good | 2 | \( 1 + (-1 + i)T - iT^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + iT^{2} \) |
| 13 | \( 1 - 2iT - T^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 31 | \( 1 + (1 + i)T + iT^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + (1 + i)T + iT^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (-1 - i)T + iT^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + 2iT - T^{2} \) |
| 61 | \( 1 - iT^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (1 - i)T - iT^{2} \) |
| 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
−9.448714402438412153078856361613, −9.013583199878180017521722825469, −7.909930291327359386067163393486, −6.82301674379719791647076645185, −5.77487939191565230315492397537, −5.06090099359527545749738494172, −4.14873997327109448197734214665, −3.85620080643315639556510783474, −2.71534346064884293610064227614, −1.80863970995744292176853033784,
1.11946198236062556720060919945, 2.76570478491314705304799496869, 3.48871735581238695571082916185, 4.84606035679535512913514622259, 5.46915395071510417297680507734, 6.08930402049363758613191143349, 7.01173845027374617510831424490, 7.38519636203980971489244978722, 8.311054210908338613961222226780, 8.818386907003194553262343213001