L(s) = 1 | + 7-s − 3·9-s + 4·11-s + 6·13-s + 2·17-s + 4·19-s − 23-s + 2·29-s + 4·31-s − 2·37-s − 6·41-s − 12·43-s − 12·47-s + 49-s − 10·53-s − 2·61-s − 3·63-s − 12·67-s − 8·71-s + 14·73-s + 4·77-s − 8·79-s + 9·81-s + 4·83-s + 6·89-s + 6·91-s + 10·97-s + ⋯ |
L(s) = 1 | + 0.377·7-s − 9-s + 1.20·11-s + 1.66·13-s + 0.485·17-s + 0.917·19-s − 0.208·23-s + 0.371·29-s + 0.718·31-s − 0.328·37-s − 0.937·41-s − 1.82·43-s − 1.75·47-s + 1/7·49-s − 1.37·53-s − 0.256·61-s − 0.377·63-s − 1.46·67-s − 0.949·71-s + 1.63·73-s + 0.455·77-s − 0.900·79-s + 81-s + 0.439·83-s + 0.635·89-s + 0.628·91-s + 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 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
−13.32283631434999, −12.43153541494961, −12.00808065713492, −11.55620066773055, −11.40710880550765, −10.90076073277481, −10.21062549155299, −9.884202594592070, −9.238842082682544, −8.789449299946218, −8.437793838850147, −8.032258681825510, −7.538707498254261, −6.688849127004371, −6.386128513942120, −6.090532042551717, −5.348616785988717, −4.986757837504405, −4.357555388519241, −3.641964096513807, −3.281462490688485, −2.964910011700232, −1.830660958507807, −1.488654244865997, −0.9316379069144031, 0,
0.9316379069144031, 1.488654244865997, 1.830660958507807, 2.964910011700232, 3.281462490688485, 3.641964096513807, 4.357555388519241, 4.986757837504405, 5.348616785988717, 6.090532042551717, 6.386128513942120, 6.688849127004371, 7.538707498254261, 8.032258681825510, 8.437793838850147, 8.789449299946218, 9.238842082682544, 9.884202594592070, 10.21062549155299, 10.90076073277481, 11.40710880550765, 11.55620066773055, 12.00808065713492, 12.43153541494961, 13.32283631434999