| L(s) = 1 | + 16·5-s + 7·7-s + 24·11-s − 68·13-s − 54·17-s + 46·19-s + 176·23-s + 131·25-s + 174·29-s + 116·31-s + 112·35-s + 74·37-s + 10·41-s + 480·43-s − 572·47-s + 49·49-s + 162·53-s + 384·55-s − 86·59-s − 904·61-s − 1.08e3·65-s − 660·67-s + 1.02e3·71-s + 770·73-s + 168·77-s + 904·79-s + 682·83-s + ⋯ |
| L(s) = 1 | + 1.43·5-s + 0.377·7-s + 0.657·11-s − 1.45·13-s − 0.770·17-s + 0.555·19-s + 1.59·23-s + 1.04·25-s + 1.11·29-s + 0.672·31-s + 0.540·35-s + 0.328·37-s + 0.0380·41-s + 1.70·43-s − 1.77·47-s + 1/7·49-s + 0.419·53-s + 0.941·55-s − 0.189·59-s − 1.89·61-s − 2.07·65-s − 1.20·67-s + 1.71·71-s + 1.23·73-s + 0.248·77-s + 1.28·79-s + 0.901·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.050178971\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.050178971\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 5 | \( 1 - 16 T + p^{3} T^{2} \) |
| 11 | \( 1 - 24 T + p^{3} T^{2} \) |
| 13 | \( 1 + 68 T + p^{3} T^{2} \) |
| 17 | \( 1 + 54 T + p^{3} T^{2} \) |
| 19 | \( 1 - 46 T + p^{3} T^{2} \) |
| 23 | \( 1 - 176 T + p^{3} T^{2} \) |
| 29 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 31 | \( 1 - 116 T + p^{3} T^{2} \) |
| 37 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 41 | \( 1 - 10 T + p^{3} T^{2} \) |
| 43 | \( 1 - 480 T + p^{3} T^{2} \) |
| 47 | \( 1 + 572 T + p^{3} T^{2} \) |
| 53 | \( 1 - 162 T + p^{3} T^{2} \) |
| 59 | \( 1 + 86 T + p^{3} T^{2} \) |
| 61 | \( 1 + 904 T + p^{3} T^{2} \) |
| 67 | \( 1 + 660 T + p^{3} T^{2} \) |
| 71 | \( 1 - 1024 T + p^{3} T^{2} \) |
| 73 | \( 1 - 770 T + p^{3} T^{2} \) |
| 79 | \( 1 - 904 T + p^{3} T^{2} \) |
| 83 | \( 1 - 682 T + p^{3} T^{2} \) |
| 89 | \( 1 - 102 T + p^{3} T^{2} \) |
| 97 | \( 1 + 218 T + p^{3} T^{2} \) |
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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
−9.423386715376178839283466413070, −9.126324745523595897671768071872, −7.892737571145305412350907699711, −6.88997934236875360947243678347, −6.24959608718530682166574476928, −5.14543658401842690186129112815, −4.60372354912900108344022658685, −2.94982907751726863014940051696, −2.10851455581435378508951746441, −0.971925103459420069850942825748,
0.971925103459420069850942825748, 2.10851455581435378508951746441, 2.94982907751726863014940051696, 4.60372354912900108344022658685, 5.14543658401842690186129112815, 6.24959608718530682166574476928, 6.88997934236875360947243678347, 7.892737571145305412350907699711, 9.126324745523595897671768071872, 9.423386715376178839283466413070
