| L(s) = 1 | + 2·3-s + 9-s + 3·11-s − 13-s + 6·17-s − 19-s + 9·23-s − 4·27-s − 6·29-s − 8·31-s + 6·33-s − 7·37-s − 2·39-s + 3·41-s − 2·43-s + 9·47-s + 12·51-s + 9·53-s − 2·57-s − 8·61-s − 8·67-s + 18·69-s + 4·73-s + 10·79-s − 11·81-s − 12·87-s + 6·89-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s + 0.904·11-s − 0.277·13-s + 1.45·17-s − 0.229·19-s + 1.87·23-s − 0.769·27-s − 1.11·29-s − 1.43·31-s + 1.04·33-s − 1.15·37-s − 0.320·39-s + 0.468·41-s − 0.304·43-s + 1.31·47-s + 1.68·51-s + 1.23·53-s − 0.264·57-s − 1.02·61-s − 0.977·67-s + 2.16·69-s + 0.468·73-s + 1.12·79-s − 1.22·81-s − 1.28·87-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.290092453\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.290092453\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.12958500676616, −13.58342001646843, −13.14784724816518, −12.52380028180095, −12.17078599073203, −11.54503444022294, −10.98133156153046, −10.50371692354634, −9.856000264051931, −9.231486325791244, −9.006643222039969, −8.669180951026093, −7.795006711890471, −7.439385330607509, −7.088522815369871, −6.318219585229804, −5.545660269027774, −5.280425774743777, −4.389555512509109, −3.666628626352585, −3.422094310318187, −2.798635713484077, −2.030737359101548, −1.471723002512010, −0.6315961530328700,
0.6315961530328700, 1.471723002512010, 2.030737359101548, 2.798635713484077, 3.422094310318187, 3.666628626352585, 4.389555512509109, 5.280425774743777, 5.545660269027774, 6.318219585229804, 7.088522815369871, 7.439385330607509, 7.795006711890471, 8.669180951026093, 9.006643222039969, 9.231486325791244, 9.856000264051931, 10.50371692354634, 10.98133156153046, 11.54503444022294, 12.17078599073203, 12.52380028180095, 13.14784724816518, 13.58342001646843, 14.12958500676616
