| L(s) = 1 | − 1.39·2-s − 0.0602·4-s + 0.392·7-s + 2.86·8-s + 2.96·11-s − 2.92·13-s − 0.547·14-s − 3.87·16-s − 0.739·17-s − 3.50·19-s − 4.12·22-s + 7.61·23-s + 4.08·26-s − 0.0236·28-s − 0.883·29-s − 31-s − 0.340·32-s + 1.03·34-s + 1.01·37-s + 4.87·38-s + 0.566·41-s − 4.98·43-s − 0.178·44-s − 10.6·46-s + 2.22·47-s − 6.84·49-s + 0.176·52-s + ⋯ |
| L(s) = 1 | − 0.984·2-s − 0.0301·4-s + 0.148·7-s + 1.01·8-s + 0.893·11-s − 0.812·13-s − 0.146·14-s − 0.968·16-s − 0.179·17-s − 0.803·19-s − 0.880·22-s + 1.58·23-s + 0.800·26-s − 0.00447·28-s − 0.164·29-s − 0.179·31-s − 0.0602·32-s + 0.176·34-s + 0.167·37-s + 0.790·38-s + 0.0885·41-s − 0.760·43-s − 0.0269·44-s − 1.56·46-s + 0.324·47-s − 0.977·49-s + 0.0244·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 1.39T + 2T^{2} \) |
| 7 | \( 1 - 0.392T + 7T^{2} \) |
| 11 | \( 1 - 2.96T + 11T^{2} \) |
| 13 | \( 1 + 2.92T + 13T^{2} \) |
| 17 | \( 1 + 0.739T + 17T^{2} \) |
| 19 | \( 1 + 3.50T + 19T^{2} \) |
| 23 | \( 1 - 7.61T + 23T^{2} \) |
| 29 | \( 1 + 0.883T + 29T^{2} \) |
| 37 | \( 1 - 1.01T + 37T^{2} \) |
| 41 | \( 1 - 0.566T + 41T^{2} \) |
| 43 | \( 1 + 4.98T + 43T^{2} \) |
| 47 | \( 1 - 2.22T + 47T^{2} \) |
| 53 | \( 1 + 7.23T + 53T^{2} \) |
| 59 | \( 1 - 4.56T + 59T^{2} \) |
| 61 | \( 1 + 2.23T + 61T^{2} \) |
| 67 | \( 1 + 3.28T + 67T^{2} \) |
| 71 | \( 1 + 9.31T + 71T^{2} \) |
| 73 | \( 1 - 15.6T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 - 5.97T + 83T^{2} \) |
| 89 | \( 1 - 2.02T + 89T^{2} \) |
| 97 | \( 1 + 2.14T + 97T^{2} \) |
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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
−7.76782064778119463795296750920, −6.96163208541234392111552100433, −6.56284259656120948029701068279, −5.39710429804550807315787042703, −4.70559957736161331074143344094, −4.09134068082977098587571384877, −3.04284070351830112303282932933, −1.97227160751131999427805007938, −1.14849271944332706875450600713, 0,
1.14849271944332706875450600713, 1.97227160751131999427805007938, 3.04284070351830112303282932933, 4.09134068082977098587571384877, 4.70559957736161331074143344094, 5.39710429804550807315787042703, 6.56284259656120948029701068279, 6.96163208541234392111552100433, 7.76782064778119463795296750920
