| L(s) = 1 | + i·2-s + 1.44i·3-s − 4-s − 1.44·6-s + 2.44i·7-s − i·8-s + 0.898·9-s + 3.44·11-s − 1.44i·12-s + 0.449i·13-s − 2.44·14-s + 16-s + 3.44i·17-s + 0.898i·18-s + 5·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.836i·3-s − 0.5·4-s − 0.591·6-s + 0.925i·7-s − 0.353i·8-s + 0.299·9-s + 1.04·11-s − 0.418i·12-s + 0.124i·13-s − 0.654·14-s + 0.250·16-s + 0.836i·17-s + 0.211i·18-s + 1.14·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.852883989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.852883989\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 37 | \( 1 + iT \) |
| good | 3 | \( 1 - 1.44iT - 3T^{2} \) |
| 7 | \( 1 - 2.44iT - 7T^{2} \) |
| 11 | \( 1 - 3.44T + 11T^{2} \) |
| 13 | \( 1 - 0.449iT - 13T^{2} \) |
| 17 | \( 1 - 3.44iT - 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 + 2iT - 23T^{2} \) |
| 29 | \( 1 - 0.898T + 29T^{2} \) |
| 31 | \( 1 - 4.44T + 31T^{2} \) |
| 41 | \( 1 - T + 41T^{2} \) |
| 43 | \( 1 - 1.10iT - 43T^{2} \) |
| 47 | \( 1 - 9.79iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 2T + 59T^{2} \) |
| 61 | \( 1 + 6.44T + 61T^{2} \) |
| 67 | \( 1 - 4.55iT - 67T^{2} \) |
| 71 | \( 1 + 7.55T + 71T^{2} \) |
| 73 | \( 1 + 12.7iT - 73T^{2} \) |
| 79 | \( 1 + 7.79T + 79T^{2} \) |
| 83 | \( 1 + 3.44iT - 83T^{2} \) |
| 89 | \( 1 + 14.3T + 89T^{2} \) |
| 97 | \( 1 - 14iT - 97T^{2} \) |
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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
−9.421546184912169820065740676765, −8.911448154250858803486337439471, −8.109997405080781646532135518523, −7.14723088954233296331321501409, −6.30102511812547621328418208401, −5.61208091827533116142243086753, −4.67623259762039360672893111546, −4.01465681252400403316205017093, −2.99770352149168965957501106108, −1.42687891893902947827277606132,
0.810763563651474057764028247760, 1.49883405044946427856726382801, 2.80258553466914138761448187447, 3.82057764802524849831635309372, 4.57602969739612288040094517818, 5.68376177975305300955621602105, 6.81432425085083373587768107409, 7.24466377257167778917351857463, 8.045317708557658968129933567554, 9.046606684330909649896165230758
