| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s + 2·13-s − 14-s + 16-s − 2·17-s + 2·19-s + 2·23-s + 2·26-s − 28-s + 4·29-s + 31-s + 32-s − 2·34-s + 3·37-s + 2·38-s − 2·41-s − 4·43-s + 2·46-s − 47-s − 6·49-s + 2·52-s − 11·53-s − 56-s + 4·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s + 0.554·13-s − 0.267·14-s + 1/4·16-s − 0.485·17-s + 0.458·19-s + 0.417·23-s + 0.392·26-s − 0.188·28-s + 0.742·29-s + 0.179·31-s + 0.176·32-s − 0.342·34-s + 0.493·37-s + 0.324·38-s − 0.312·41-s − 0.609·43-s + 0.294·46-s − 0.145·47-s − 6/7·49-s + 0.277·52-s − 1.51·53-s − 0.133·56-s + 0.525·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 167 | \( 1 + T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−14.23414444105645, −13.79194903722925, −13.29950729729166, −12.87492667232042, −12.47390936017634, −11.78654956633383, −11.40504161724156, −10.95380636574190, −10.34331895438444, −9.847396798891196, −9.308272784182983, −8.693079023090246, −8.160236477952573, −7.613265649840013, −6.935243956468317, −6.472976909701590, −6.092610961750576, −5.379766482421044, −4.820709914296853, −4.347610114464212, −3.603453570134553, −3.120235754822049, −2.578506346847799, −1.710698527026169, −1.069093644039577, 0,
1.069093644039577, 1.710698527026169, 2.578506346847799, 3.120235754822049, 3.603453570134553, 4.347610114464212, 4.820709914296853, 5.379766482421044, 6.092610961750576, 6.472976909701590, 6.935243956468317, 7.613265649840013, 8.160236477952573, 8.693079023090246, 9.308272784182983, 9.847396798891196, 10.34331895438444, 10.95380636574190, 11.40504161724156, 11.78654956633383, 12.47390936017634, 12.87492667232042, 13.29950729729166, 13.79194903722925, 14.23414444105645
