| L(s) = 1 | + 7-s − 4·11-s − 2·13-s − 6·17-s − 8·23-s + 10·29-s − 8·31-s + 2·37-s + 2·41-s + 8·43-s + 4·47-s + 49-s − 10·53-s + 4·59-s + 6·61-s + 12·71-s + 6·73-s − 4·77-s − 8·79-s + 4·83-s − 14·89-s − 2·91-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 1.20·11-s − 0.554·13-s − 1.45·17-s − 1.66·23-s + 1.85·29-s − 1.43·31-s + 0.328·37-s + 0.312·41-s + 1.21·43-s + 0.583·47-s + 1/7·49-s − 1.37·53-s + 0.520·59-s + 0.768·61-s + 1.42·71-s + 0.702·73-s − 0.455·77-s − 0.900·79-s + 0.439·83-s − 1.48·89-s − 0.209·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.01132859468960, −13.60689674897565, −12.84268920730867, −12.63768245830831, −12.14137442617780, −11.43278799452436, −11.03292917933179, −10.63430980738514, −10.02357429473196, −9.674126469342024, −8.973774093280819, −8.430250929551331, −8.062732431584275, −7.481023067248277, −7.054263852533721, −6.326960167848495, −5.891725928282799, −5.230017109722563, −4.725441451936100, −4.260766929979407, −3.641398596631973, −2.706915300738555, −2.345117557296558, −1.825956738134954, −0.7275045590186833, 0,
0.7275045590186833, 1.825956738134954, 2.345117557296558, 2.706915300738555, 3.641398596631973, 4.260766929979407, 4.725441451936100, 5.230017109722563, 5.891725928282799, 6.326960167848495, 7.054263852533721, 7.481023067248277, 8.062732431584275, 8.430250929551331, 8.973774093280819, 9.674126469342024, 10.02357429473196, 10.63430980738514, 11.03292917933179, 11.43278799452436, 12.14137442617780, 12.63768245830831, 12.84268920730867, 13.60689674897565, 14.01132859468960
