| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 4.77·11-s + 2.77·13-s − 14-s + 16-s − 4.77·17-s + 2·19-s − 4.77·22-s + 4.77·23-s + 2.77·26-s − 28-s − 3·29-s + 8.54·31-s + 32-s − 4.77·34-s − 0.227·37-s + 2·38-s + 1.77·41-s − 6.77·43-s − 4.77·44-s + 4.77·46-s − 12.5·47-s + 49-s + 2.77·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.377·7-s + 0.353·8-s − 1.43·11-s + 0.768·13-s − 0.267·14-s + 0.250·16-s − 1.15·17-s + 0.458·19-s − 1.01·22-s + 0.995·23-s + 0.543·26-s − 0.188·28-s − 0.557·29-s + 1.53·31-s + 0.176·32-s − 0.818·34-s − 0.0374·37-s + 0.324·38-s + 0.276·41-s − 1.03·43-s − 0.719·44-s + 0.703·46-s − 1.82·47-s + 0.142·49-s + 0.384·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 4.77T + 11T^{2} \) |
| 13 | \( 1 - 2.77T + 13T^{2} \) |
| 17 | \( 1 + 4.77T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 4.77T + 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 - 8.54T + 31T^{2} \) |
| 37 | \( 1 + 0.227T + 37T^{2} \) |
| 41 | \( 1 - 1.77T + 41T^{2} \) |
| 43 | \( 1 + 6.77T + 43T^{2} \) |
| 47 | \( 1 + 12.5T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 1.77T + 59T^{2} \) |
| 61 | \( 1 + 5.77T + 61T^{2} \) |
| 67 | \( 1 - 7.54T + 67T^{2} \) |
| 71 | \( 1 - 1.77T + 71T^{2} \) |
| 73 | \( 1 + 6.22T + 73T^{2} \) |
| 79 | \( 1 - 6.77T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 3.54T + 89T^{2} \) |
| 97 | \( 1 - 0.455T + 97T^{2} \) |
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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
−7.22855464936122971345524074469, −6.52757488701487715872372887834, −6.05465395241947365589568937415, −5.06046951297725177972500955992, −4.81374014918243590071184090877, −3.77429351946961755296864143686, −3.03842820994543051508748347872, −2.46484523349440843248370733036, −1.36673397344874775275617729420, 0,
1.36673397344874775275617729420, 2.46484523349440843248370733036, 3.03842820994543051508748347872, 3.77429351946961755296864143686, 4.81374014918243590071184090877, 5.06046951297725177972500955992, 6.05465395241947365589568937415, 6.52757488701487715872372887834, 7.22855464936122971345524074469
