| L(s) = 1 | − 3-s − 5-s + 7-s + 9-s + 6·13-s + 15-s + 2·17-s + 8·19-s − 21-s + 8·23-s + 25-s − 27-s + 2·29-s + 4·31-s − 35-s + 2·37-s − 6·39-s − 6·41-s − 4·43-s − 45-s + 8·47-s + 49-s − 2·51-s − 10·53-s − 8·57-s − 4·59-s + 2·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.66·13-s + 0.258·15-s + 0.485·17-s + 1.83·19-s − 0.218·21-s + 1.66·23-s + 1/5·25-s − 0.192·27-s + 0.371·29-s + 0.718·31-s − 0.169·35-s + 0.328·37-s − 0.960·39-s − 0.937·41-s − 0.609·43-s − 0.149·45-s + 1.16·47-s + 1/7·49-s − 0.280·51-s − 1.37·53-s − 1.05·57-s − 0.520·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.073043849\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.073043849\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−7.890343804515539767082945575168, −7.31064486223013717817840097287, −6.56081791134764681896703052094, −5.83809715420811320017380787786, −5.14166505548978298646598195469, −4.51866305684229912467918947892, −3.49820192069350688210369187396, −3.01341389876459207175811761265, −1.42099904573672018408228849546, −0.885814180804581930260732198060,
0.885814180804581930260732198060, 1.42099904573672018408228849546, 3.01341389876459207175811761265, 3.49820192069350688210369187396, 4.51866305684229912467918947892, 5.14166505548978298646598195469, 5.83809715420811320017380787786, 6.56081791134764681896703052094, 7.31064486223013717817840097287, 7.890343804515539767082945575168
