| L(s) = 1 | + 2·7-s − 3·9-s − 6·13-s − 4·17-s + 4·19-s + 6·23-s − 6·29-s + 8·31-s − 8·37-s + 6·41-s + 10·43-s − 3·49-s + 12·53-s − 8·59-s − 61-s − 6·63-s − 2·67-s − 12·71-s − 14·73-s + 16·79-s + 9·81-s + 8·83-s + 6·89-s − 12·91-s − 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 9-s − 1.66·13-s − 0.970·17-s + 0.917·19-s + 1.25·23-s − 1.11·29-s + 1.43·31-s − 1.31·37-s + 0.937·41-s + 1.52·43-s − 3/7·49-s + 1.64·53-s − 1.04·59-s − 0.128·61-s − 0.755·63-s − 0.244·67-s − 1.42·71-s − 1.63·73-s + 1.80·79-s + 81-s + 0.878·83-s + 0.635·89-s − 1.25·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 61 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−15.57162026308500, −14.97487219178676, −14.66024264632138, −14.15119011278188, −13.55921578285505, −13.09214478121413, −12.14605264706524, −11.99858621055257, −11.34982554276780, −10.82354593790546, −10.33841119615049, −9.439404527902631, −9.109162142856839, −8.569392157787636, −7.718646756625312, −7.439120011024547, −6.782361255391139, −5.947536393722238, −5.327503680631580, −4.844825792817033, −4.305531570902132, −3.260321016385096, −2.636660954597023, −2.091050287930552, −0.9984320753813225, 0,
0.9984320753813225, 2.091050287930552, 2.636660954597023, 3.260321016385096, 4.305531570902132, 4.844825792817033, 5.327503680631580, 5.947536393722238, 6.782361255391139, 7.439120011024547, 7.718646756625312, 8.569392157787636, 9.109162142856839, 9.439404527902631, 10.33841119615049, 10.82354593790546, 11.34982554276780, 11.99858621055257, 12.14605264706524, 13.09214478121413, 13.55921578285505, 14.15119011278188, 14.66024264632138, 14.97487219178676, 15.57162026308500
