L(s) = 1 | − 3-s + 9-s − 4·11-s + 6·13-s − 6·17-s + 4·19-s − 8·23-s − 27-s + 10·29-s + 4·31-s + 4·33-s + 6·37-s − 6·39-s − 6·41-s + 4·43-s + 12·47-s + 6·51-s − 6·53-s − 4·57-s + 4·59-s + 2·61-s + 4·67-s + 8·69-s − 2·73-s + 8·79-s + 81-s + 12·83-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.20·11-s + 1.66·13-s − 1.45·17-s + 0.917·19-s − 1.66·23-s − 0.192·27-s + 1.85·29-s + 0.718·31-s + 0.696·33-s + 0.986·37-s − 0.960·39-s − 0.937·41-s + 0.609·43-s + 1.75·47-s + 0.840·51-s − 0.824·53-s − 0.529·57-s + 0.520·59-s + 0.256·61-s + 0.488·67-s + 0.963·69-s − 0.234·73-s + 0.900·79-s + 1/9·81-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.934530729\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.934530729\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−13.55330186392606, −13.19112760211629, −12.64609798697480, −11.99502658603166, −11.72244455993245, −11.08686562692144, −10.70489942461262, −10.30339020629031, −9.834546229501197, −9.141935976128729, −8.617474229941711, −8.036967333840397, −7.867363700307072, −6.941543956774673, −6.527313935390201, −6.019047659378486, −5.631884833984712, −4.941085462658048, −4.393623621527073, −3.960082281066867, −3.187267088723402, −2.538290894256664, −1.971501226669912, −1.052451844894350, −0.5108284773752048,
0.5108284773752048, 1.052451844894350, 1.971501226669912, 2.538290894256664, 3.187267088723402, 3.960082281066867, 4.393623621527073, 4.941085462658048, 5.631884833984712, 6.019047659378486, 6.527313935390201, 6.941543956774673, 7.867363700307072, 8.036967333840397, 8.617474229941711, 9.141935976128729, 9.834546229501197, 10.30339020629031, 10.70489942461262, 11.08686562692144, 11.72244455993245, 11.99502658603166, 12.64609798697480, 13.19112760211629, 13.55330186392606