| L(s) = 1 | − 2-s + 4-s + 3·7-s − 8-s − 11-s + 2·13-s − 3·14-s + 16-s − 17-s − 7·19-s + 22-s + 9·23-s − 2·26-s + 3·28-s + 6·29-s − 10·31-s − 32-s + 34-s − 4·37-s + 7·38-s + 3·41-s − 12·43-s − 44-s − 9·46-s + 10·47-s + 2·49-s + 2·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.13·7-s − 0.353·8-s − 0.301·11-s + 0.554·13-s − 0.801·14-s + 1/4·16-s − 0.242·17-s − 1.60·19-s + 0.213·22-s + 1.87·23-s − 0.392·26-s + 0.566·28-s + 1.11·29-s − 1.79·31-s − 0.176·32-s + 0.171·34-s − 0.657·37-s + 1.13·38-s + 0.468·41-s − 1.82·43-s − 0.150·44-s − 1.32·46-s + 1.45·47-s + 2/7·49-s + 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 127 | \( 1 - T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−14.72343614798765, −14.36920176112989, −13.42936838731552, −13.19192018147048, −12.51961532757206, −12.03720939129475, −11.29359957184171, −10.94475133554789, −10.70940397151679, −10.11062144605062, −9.315110515225903, −8.788159556249965, −8.490057630269620, −8.030674865902080, −7.322822043309637, −6.801928336746932, −6.400810998552542, −5.408256408603296, −5.176147650431908, −4.384347035634956, −3.786491064582191, −2.955859029855544, −2.261977495305240, −1.654076996752007, −0.9956493611761336, 0,
0.9956493611761336, 1.654076996752007, 2.261977495305240, 2.955859029855544, 3.786491064582191, 4.384347035634956, 5.176147650431908, 5.408256408603296, 6.400810998552542, 6.801928336746932, 7.322822043309637, 8.030674865902080, 8.490057630269620, 8.788159556249965, 9.315110515225903, 10.11062144605062, 10.70940397151679, 10.94475133554789, 11.29359957184171, 12.03720939129475, 12.51961532757206, 13.19192018147048, 13.42936838731552, 14.36920176112989, 14.72343614798765
