| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 7-s + 8-s + 9-s − 12-s + 13-s + 14-s + 16-s + 17-s + 18-s − 19-s − 21-s − 23-s − 24-s + 26-s − 27-s + 28-s − 29-s + 31-s + 32-s + 34-s + 36-s − 37-s − 38-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 7-s + 8-s + 9-s − 12-s + 13-s + 14-s + 16-s + 17-s + 18-s − 19-s − 21-s − 23-s − 24-s + 26-s − 27-s + 28-s − 29-s + 31-s + 32-s + 34-s + 36-s − 37-s − 38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.431063069\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.431063069\) |
| \(L(1)\) |
\(\approx\) |
\(1.694449067\) |
| \(L(1)\) |
\(\approx\) |
\(1.694449067\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 - T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−32.992676266835819663823465156577, −31.802647200809233646907352040814, −30.36075907982639052006903792501, −29.86220196587676814080729191581, −28.393839274654861558429346263102, −27.61700479243725250676142644445, −25.796150909371293478307837457034, −24.38737830488568094702011150827, −23.60986162844672312553194126411, −22.68102708815235806998329686677, −21.41879925718685876264250429310, −20.71867097669620387938655234510, −18.851541393180054554974977118526, −17.44785743430645892914024382800, −16.30264279220135006390959232093, −15.108776587227819884423539999057, −13.77348440625003556944408026897, −12.40471443568932653648515943841, −11.40043623969198003422829975849, −10.42315246404025260876883624454, −7.92367740819724180850066627351, −6.37005874038062820681872530606, −5.25631708635296194324745449220, −3.97704030111212558605523918635, −1.59805935779877015404751613070,
1.59805935779877015404751613070, 3.97704030111212558605523918635, 5.25631708635296194324745449220, 6.37005874038062820681872530606, 7.92367740819724180850066627351, 10.42315246404025260876883624454, 11.40043623969198003422829975849, 12.40471443568932653648515943841, 13.77348440625003556944408026897, 15.108776587227819884423539999057, 16.30264279220135006390959232093, 17.44785743430645892914024382800, 18.851541393180054554974977118526, 20.71867097669620387938655234510, 21.41879925718685876264250429310, 22.68102708815235806998329686677, 23.60986162844672312553194126411, 24.38737830488568094702011150827, 25.796150909371293478307837457034, 27.61700479243725250676142644445, 28.393839274654861558429346263102, 29.86220196587676814080729191581, 30.36075907982639052006903792501, 31.802647200809233646907352040814, 32.992676266835819663823465156577
