| L(s) = 1 | + 5·11-s − 2·13-s − 6·17-s − 2·19-s − 5·23-s + 5·29-s − 4·31-s + 37-s + 12·41-s + 5·43-s − 2·47-s − 14·53-s − 2·59-s − 5·67-s + 9·71-s + 10·73-s + 11·79-s − 16·83-s + 14·89-s + 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 1.50·11-s − 0.554·13-s − 1.45·17-s − 0.458·19-s − 1.04·23-s + 0.928·29-s − 0.718·31-s + 0.164·37-s + 1.87·41-s + 0.762·43-s − 0.291·47-s − 1.92·53-s − 0.260·59-s − 0.610·67-s + 1.06·71-s + 1.17·73-s + 1.23·79-s − 1.75·83-s + 1.48·89-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.827014192\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.827014192\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−14.10780248807980, −13.42346017550523, −12.80006549543623, −12.40469923229188, −11.98439076885809, −11.39219610182622, −10.91388815725765, −10.59482680583682, −9.697587856006791, −9.330650378184544, −9.098398037426611, −8.280394478016344, −7.944856872834665, −7.182843484480551, −6.686783699717013, −6.265753561990724, −5.833990856626149, −4.922855833481243, −4.387249550852844, −4.070877314060662, −3.378920259659067, −2.513510446812525, −2.055935711153675, −1.319703039784504, −0.4382396574354547,
0.4382396574354547, 1.319703039784504, 2.055935711153675, 2.513510446812525, 3.378920259659067, 4.070877314060662, 4.387249550852844, 4.922855833481243, 5.833990856626149, 6.265753561990724, 6.686783699717013, 7.182843484480551, 7.944856872834665, 8.280394478016344, 9.098398037426611, 9.330650378184544, 9.697587856006791, 10.59482680583682, 10.91388815725765, 11.39219610182622, 11.98439076885809, 12.40469923229188, 12.80006549543623, 13.42346017550523, 14.10780248807980
