L(s) = 1 | + 2-s + 4-s + 2·7-s + 8-s − 6·13-s + 2·14-s + 16-s + 8·17-s + 19-s − 4·23-s − 6·26-s + 2·28-s − 2·29-s − 2·31-s + 32-s + 8·34-s + 2·37-s + 38-s + 12·41-s − 4·43-s − 4·46-s + 12·47-s − 3·49-s − 6·52-s + 10·53-s + 2·56-s − 2·58-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.755·7-s + 0.353·8-s − 1.66·13-s + 0.534·14-s + 1/4·16-s + 1.94·17-s + 0.229·19-s − 0.834·23-s − 1.17·26-s + 0.377·28-s − 0.371·29-s − 0.359·31-s + 0.176·32-s + 1.37·34-s + 0.328·37-s + 0.162·38-s + 1.87·41-s − 0.609·43-s − 0.589·46-s + 1.75·47-s − 3/7·49-s − 0.832·52-s + 1.37·53-s + 0.267·56-s − 0.262·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.660383535\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.660383535\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.69415048487312640020695649806, −7.26198939744952637986431277341, −6.24412309320927881624133826003, −5.48209761018056212659943996496, −5.11803505906494952436113350084, −4.29969803744425454809856970718, −3.57858037003784891705947799514, −2.65348829938804265663919071891, −1.97067397788643589050294424348, −0.847955580652318344970204522998,
0.847955580652318344970204522998, 1.97067397788643589050294424348, 2.65348829938804265663919071891, 3.57858037003784891705947799514, 4.29969803744425454809856970718, 5.11803505906494952436113350084, 5.48209761018056212659943996496, 6.24412309320927881624133826003, 7.26198939744952637986431277341, 7.69415048487312640020695649806