L(s) = 1 | + 3-s + 7-s + 9-s + 11-s + 4·13-s − 4·17-s + 4·19-s + 21-s + 6·23-s + 27-s + 10·29-s + 33-s + 8·37-s + 4·39-s + 6·41-s − 8·47-s + 49-s − 4·51-s − 10·53-s + 4·57-s − 8·59-s − 10·61-s + 63-s + 14·67-s + 6·69-s − 12·71-s + 10·73-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.301·11-s + 1.10·13-s − 0.970·17-s + 0.917·19-s + 0.218·21-s + 1.25·23-s + 0.192·27-s + 1.85·29-s + 0.174·33-s + 1.31·37-s + 0.640·39-s + 0.937·41-s − 1.16·47-s + 1/7·49-s − 0.560·51-s − 1.37·53-s + 0.529·57-s − 1.04·59-s − 1.28·61-s + 0.125·63-s + 1.71·67-s + 0.722·69-s − 1.42·71-s + 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.323007368\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.323007368\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−14.54703839841363, −13.99294883032507, −13.74697247485950, −13.11162352526724, −12.69360211188991, −12.04030536688141, −11.40726152932767, −10.95235044947565, −10.65613984616373, −9.674950520256829, −9.379388018471422, −8.850199833625178, −8.193300718026115, −7.946669136868100, −7.142807068289150, −6.538431070680204, −6.184766267014937, −5.314656443298157, −4.572483132382957, −4.341658722708331, −3.286449078068067, −3.034228164649271, −2.159818472754163, −1.331284550802149, −0.7946989408295984,
0.7946989408295984, 1.331284550802149, 2.159818472754163, 3.034228164649271, 3.286449078068067, 4.341658722708331, 4.572483132382957, 5.314656443298157, 6.184766267014937, 6.538431070680204, 7.142807068289150, 7.946669136868100, 8.193300718026115, 8.850199833625178, 9.379388018471422, 9.674950520256829, 10.65613984616373, 10.95235044947565, 11.40726152932767, 12.04030536688141, 12.69360211188991, 13.11162352526724, 13.74697247485950, 13.99294883032507, 14.54703839841363