| L(s) = 1 | + 4·7-s − 4·11-s + 2·13-s − 2·17-s + 19-s + 8·23-s − 6·29-s − 4·31-s + 10·37-s + 2·41-s + 12·43-s + 9·49-s + 6·53-s − 10·61-s − 4·67-s − 8·71-s − 2·73-s − 16·77-s + 12·79-s + 8·83-s − 6·89-s + 8·91-s − 18·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 1.20·11-s + 0.554·13-s − 0.485·17-s + 0.229·19-s + 1.66·23-s − 1.11·29-s − 0.718·31-s + 1.64·37-s + 0.312·41-s + 1.82·43-s + 9/7·49-s + 0.824·53-s − 1.28·61-s − 0.488·67-s − 0.949·71-s − 0.234·73-s − 1.82·77-s + 1.35·79-s + 0.878·83-s − 0.635·89-s + 0.838·91-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.099416672\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.099416672\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−14.23259026192142, −13.57787130199162, −13.19285306718509, −12.78642100168706, −12.18068534458413, −11.36916677468766, −11.14698687918473, −10.81868556558391, −10.33351546974347, −9.408208726575206, −9.057217964095989, −8.544441197043719, −7.892467391904119, −7.481291602954575, −7.218167008838212, −6.175465749641739, −5.701123489655700, −5.180073484634664, −4.636847566927571, −4.191248391352113, −3.336069038172037, −2.621910066479478, −2.090763883835203, −1.308529548107153, −0.6216019175019940,
0.6216019175019940, 1.308529548107153, 2.090763883835203, 2.621910066479478, 3.336069038172037, 4.191248391352113, 4.636847566927571, 5.180073484634664, 5.701123489655700, 6.175465749641739, 7.218167008838212, 7.481291602954575, 7.892467391904119, 8.544441197043719, 9.057217964095989, 9.408208726575206, 10.33351546974347, 10.81868556558391, 11.14698687918473, 11.36916677468766, 12.18068534458413, 12.78642100168706, 13.19285306718509, 13.57787130199162, 14.23259026192142
