| L(s) = 1 | − 2-s − 4-s + 7-s + 3·8-s + 6·13-s − 14-s − 16-s + 4·19-s + 2·23-s − 6·26-s − 28-s − 29-s + 8·31-s − 5·32-s + 3·37-s − 4·38-s − 10·41-s − 10·43-s − 2·46-s + 7·47-s + 49-s − 6·52-s − 6·53-s + 3·56-s + 58-s − 3·59-s − 6·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.377·7-s + 1.06·8-s + 1.66·13-s − 0.267·14-s − 1/4·16-s + 0.917·19-s + 0.417·23-s − 1.17·26-s − 0.188·28-s − 0.185·29-s + 1.43·31-s − 0.883·32-s + 0.493·37-s − 0.648·38-s − 1.56·41-s − 1.52·43-s − 0.294·46-s + 1.02·47-s + 1/7·49-s − 0.832·52-s − 0.824·53-s + 0.400·56-s + 0.131·58-s − 0.390·59-s − 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−13.42871059909785, −13.12974325104884, −12.30772664041838, −11.79392814763814, −11.46206758017840, −10.85569192131417, −10.46873533154557, −10.03162602375899, −9.541793137391026, −8.896302464206776, −8.714133935509066, −8.155349272595160, −7.812714273577563, −7.210399006265977, −6.667319230349604, −6.048173475857662, −5.605080052391895, −4.870781597813575, −4.590599809591324, −3.915361329011656, −3.327157374888655, −2.881201253904295, −1.791125273978863, −1.368660417436610, −0.8908224319639783, 0,
0.8908224319639783, 1.368660417436610, 1.791125273978863, 2.881201253904295, 3.327157374888655, 3.915361329011656, 4.590599809591324, 4.870781597813575, 5.605080052391895, 6.048173475857662, 6.667319230349604, 7.210399006265977, 7.812714273577563, 8.155349272595160, 8.714133935509066, 8.896302464206776, 9.541793137391026, 10.03162602375899, 10.46873533154557, 10.85569192131417, 11.46206758017840, 11.79392814763814, 12.30772664041838, 13.12974325104884, 13.42871059909785
