| L(s) = 1 | − 2·2-s + 2·4-s − 16·7-s − 8·11-s + 6·13-s + 32·14-s − 4·16-s + 38·17-s + 16·22-s − 40·23-s − 12·26-s − 32·28-s − 88·31-s + 8·32-s − 76·34-s + 6·37-s − 140·41-s − 72·43-s − 16·44-s + 80·46-s + 128·49-s + 12·52-s + 34·53-s + 144·61-s + 176·62-s − 8·64-s − 88·67-s + ⋯ |
| L(s) = 1 | − 2-s + 1/2·4-s − 2.28·7-s − 0.727·11-s + 6/13·13-s + 16/7·14-s − 1/4·16-s + 2.23·17-s + 8/11·22-s − 1.73·23-s − 0.461·26-s − 8/7·28-s − 2.83·31-s + 1/4·32-s − 2.23·34-s + 6/37·37-s − 3.41·41-s − 1.67·43-s − 0.363·44-s + 1.73·46-s + 2.61·49-s + 3/13·52-s + 0.641·53-s + 2.36·61-s + 2.83·62-s − 1/8·64-s − 1.31·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.07392350324\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07392350324\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 16 T + 128 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 38 T + 722 T^{2} - 38 p^{2} T^{3} + p^{4} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 658 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 40 T + 800 T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 238 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 44 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 70 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 72 T + 2592 T^{2} + 72 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 90 T + p^{2} T^{2} )( 1 + 56 T + p^{2} T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 1502 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 72 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 88 T + 3872 T^{2} + 88 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 88 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 110 T + 6050 T^{2} + 110 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 12338 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 48 T + 1152 T^{2} + 48 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 15166 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 114 T + 6498 T^{2} - 114 p^{2} T^{3} + p^{4} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.83966522828955670631599748696, −10.23082141797258552744902082834, −10.22034624286795423386380823220, −9.696483258161712578888244680512, −9.626845621342310559552258574870, −8.832867807972033874485261016209, −8.469282144707650670681721653000, −7.961506920120664161149201375516, −7.53053320659140939692764005735, −6.86264598394678869840106784632, −6.70656829832028699702091123216, −5.97320196123695099718663646395, −5.48105698030754757602020874427, −5.21161925466926793966756956308, −3.85136565394439117212369935246, −3.50729865234556249462843675847, −3.23804698288568186642212560334, −2.23179170563091479730648290171, −1.42004318199559738788500322000, −0.13518218753306308676935631220,
0.13518218753306308676935631220, 1.42004318199559738788500322000, 2.23179170563091479730648290171, 3.23804698288568186642212560334, 3.50729865234556249462843675847, 3.85136565394439117212369935246, 5.21161925466926793966756956308, 5.48105698030754757602020874427, 5.97320196123695099718663646395, 6.70656829832028699702091123216, 6.86264598394678869840106784632, 7.53053320659140939692764005735, 7.961506920120664161149201375516, 8.469282144707650670681721653000, 8.832867807972033874485261016209, 9.626845621342310559552258574870, 9.696483258161712578888244680512, 10.22034624286795423386380823220, 10.23082141797258552744902082834, 10.83966522828955670631599748696
