| L(s) = 1 | − 4·7-s − 4·11-s + 4·13-s − 6·17-s − 4·19-s + 4·23-s + 4·29-s + 4·37-s + 8·41-s + 12·47-s + 9·49-s + 2·53-s + 12·59-s − 2·61-s − 8·67-s − 8·71-s + 16·73-s + 16·77-s + 8·79-s − 8·83-s − 16·91-s + 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 1.20·11-s + 1.10·13-s − 1.45·17-s − 0.917·19-s + 0.834·23-s + 0.742·29-s + 0.657·37-s + 1.24·41-s + 1.75·47-s + 9/7·49-s + 0.274·53-s + 1.56·59-s − 0.256·61-s − 0.977·67-s − 0.949·71-s + 1.87·73-s + 1.82·77-s + 0.900·79-s − 0.878·83-s − 1.67·91-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−16.22684173664300, −15.84327071072423, −15.43764029277086, −14.93196339445897, −13.96657118967984, −13.39460589457839, −13.04098512527774, −12.74926564567455, −11.97196997026207, −11.05749431839502, −10.73692893144973, −10.25014230780563, −9.445278422996536, −8.908377843491971, −8.473715446527677, −7.622066475811563, −6.919309262225814, −6.348460075863729, −5.941278153417581, −5.081381302639311, −4.228317013351631, −3.689808678582719, −2.663868383915525, −2.438136360902044, −0.9415531988633392, 0,
0.9415531988633392, 2.438136360902044, 2.663868383915525, 3.689808678582719, 4.228317013351631, 5.081381302639311, 5.941278153417581, 6.348460075863729, 6.919309262225814, 7.622066475811563, 8.473715446527677, 8.908377843491971, 9.445278422996536, 10.25014230780563, 10.73692893144973, 11.05749431839502, 11.97196997026207, 12.74926564567455, 13.04098512527774, 13.39460589457839, 13.96657118967984, 14.93196339445897, 15.43764029277086, 15.84327071072423, 16.22684173664300
