L(s) = 1 | + 0.618·3-s + 7-s − 0.618·9-s + i·11-s + 0.618i·13-s + i·17-s − 0.618i·19-s + 0.618·21-s − 27-s + 29-s + 1.61i·31-s + 0.618i·33-s + 0.381i·39-s + 41-s − 43-s + ⋯ |
L(s) = 1 | + 0.618·3-s + 7-s − 0.618·9-s + i·11-s + 0.618i·13-s + i·17-s − 0.618i·19-s + 0.618·21-s − 27-s + 29-s + 1.61i·31-s + 0.618i·33-s + 0.381i·39-s + 41-s − 43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.665965829\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.665965829\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - 0.618T + T^{2} \) |
| 7 | \( 1 - T + T^{2} \) |
| 11 | \( 1 - iT - T^{2} \) |
| 13 | \( 1 - 0.618iT - T^{2} \) |
| 17 | \( 1 - iT - T^{2} \) |
| 19 | \( 1 + 0.618iT - T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( 1 - 1.61iT - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 - 1.61T + T^{2} \) |
| 53 | \( 1 + 1.61iT - T^{2} \) |
| 59 | \( 1 - 1.61iT - T^{2} \) |
| 61 | \( 1 - 0.618T + T^{2} \) |
| 67 | \( 1 + 1.61T + T^{2} \) |
| 71 | \( 1 - iT - T^{2} \) |
| 73 | \( 1 + 1.61iT - T^{2} \) |
| 79 | \( 1 + iT - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + 1.61iT - T^{2} \) |
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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
−8.740188626618862092852198141463, −8.090316399249803900549980951911, −7.35896025372094302524491186733, −6.65171439166113943830585481667, −5.70196599494598929383165716967, −4.81249691252896212866574397069, −4.26551612399996911637692952806, −3.20684500268021427538012663884, −2.26584819545634445180300634459, −1.51214057368935250304377143759,
0.909903505721793999753753277241, 2.28672155570065675977511894169, 2.95431098526613991762991281851, 3.84446131860016018738678270684, 4.78882258062993015213640737377, 5.62084944833434498516951877002, 6.14249463876937981513856908204, 7.37289842571291583557035392842, 7.965084578149038080305903315515, 8.405912234660421703102026341689