| L(s) = 1 | + 3-s − 5-s + 9-s − 4·11-s + 6·13-s − 15-s − 6·17-s + 4·19-s − 4·23-s + 25-s + 27-s − 2·29-s − 8·31-s − 4·33-s + 6·37-s + 6·39-s − 6·41-s + 8·43-s − 45-s − 6·51-s + 6·53-s + 4·55-s + 4·57-s + 4·59-s − 10·61-s − 6·65-s + 8·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s + 1.66·13-s − 0.258·15-s − 1.45·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 1.43·31-s − 0.696·33-s + 0.986·37-s + 0.960·39-s − 0.937·41-s + 1.21·43-s − 0.149·45-s − 0.840·51-s + 0.824·53-s + 0.539·55-s + 0.529·57-s + 0.520·59-s − 1.28·61-s − 0.744·65-s + 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.010734205\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.010734205\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 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 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−15.47749804926022, −15.14103478837543, −14.25444861346691, −13.75558682119861, −13.37510740271438, −12.85145122755669, −12.37318771586524, −11.41243572727907, −11.10264105175249, −10.67430409464164, −9.917365352108784, −9.247778948908961, −8.788470168391582, −8.117340426071700, −7.828721528548206, −7.072023830499384, −6.490691625267554, −5.673526840235439, −5.211462190250738, −4.172114029485778, −3.886857346080815, −3.086121523012512, −2.376363550536300, −1.626101115772469, −0.5431330312507357,
0.5431330312507357, 1.626101115772469, 2.376363550536300, 3.086121523012512, 3.886857346080815, 4.172114029485778, 5.211462190250738, 5.673526840235439, 6.490691625267554, 7.072023830499384, 7.828721528548206, 8.117340426071700, 8.788470168391582, 9.247778948908961, 9.917365352108784, 10.67430409464164, 11.10264105175249, 11.41243572727907, 12.37318771586524, 12.85145122755669, 13.37510740271438, 13.75558682119861, 14.25444861346691, 15.14103478837543, 15.47749804926022
