| L(s) = 1 | + 3-s + 5-s − 4.84·7-s + 9-s + 2.99·11-s + 3.60·13-s + 15-s + 2.27·17-s − 7.97·19-s − 4.84·21-s − 5.80·23-s + 25-s + 27-s + 2.96·29-s + 4.58·31-s + 2.99·33-s − 4.84·35-s + 9.44·37-s + 3.60·39-s − 1.48·41-s + 10.9·43-s + 45-s − 0.843·47-s + 16.4·49-s + 2.27·51-s + 5.10·53-s + 2.99·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.83·7-s + 0.333·9-s + 0.901·11-s + 1.00·13-s + 0.258·15-s + 0.550·17-s − 1.82·19-s − 1.05·21-s − 1.20·23-s + 0.200·25-s + 0.192·27-s + 0.549·29-s + 0.824·31-s + 0.520·33-s − 0.818·35-s + 1.55·37-s + 0.577·39-s − 0.231·41-s + 1.66·43-s + 0.149·45-s − 0.123·47-s + 2.34·49-s + 0.317·51-s + 0.701·53-s + 0.403·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.192426607\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.192426607\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| good | 7 | \( 1 + 4.84T + 7T^{2} \) |
| 11 | \( 1 - 2.99T + 11T^{2} \) |
| 13 | \( 1 - 3.60T + 13T^{2} \) |
| 17 | \( 1 - 2.27T + 17T^{2} \) |
| 19 | \( 1 + 7.97T + 19T^{2} \) |
| 23 | \( 1 + 5.80T + 23T^{2} \) |
| 29 | \( 1 - 2.96T + 29T^{2} \) |
| 31 | \( 1 - 4.58T + 31T^{2} \) |
| 37 | \( 1 - 9.44T + 37T^{2} \) |
| 41 | \( 1 + 1.48T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 + 0.843T + 47T^{2} \) |
| 53 | \( 1 - 5.10T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 - 10.2T + 61T^{2} \) |
| 71 | \( 1 - 1.79T + 71T^{2} \) |
| 73 | \( 1 - 2.50T + 73T^{2} \) |
| 79 | \( 1 - 5.85T + 79T^{2} \) |
| 83 | \( 1 + 0.213T + 83T^{2} \) |
| 89 | \( 1 + 2.60T + 89T^{2} \) |
| 97 | \( 1 + 10.5T + 97T^{2} \) |
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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
−8.629919255911967737396793915556, −7.79921430351400031936651109416, −6.72092981790788441172025917740, −6.25852308531426115912748548254, −5.90318313042042322142768033797, −4.29627768135803063605852866387, −3.84585025007864591564218544625, −2.96412321183185493433033126292, −2.14237067573081077624282783612, −0.822889663965966532829509629722,
0.822889663965966532829509629722, 2.14237067573081077624282783612, 2.96412321183185493433033126292, 3.84585025007864591564218544625, 4.29627768135803063605852866387, 5.90318313042042322142768033797, 6.25852308531426115912748548254, 6.72092981790788441172025917740, 7.79921430351400031936651109416, 8.629919255911967737396793915556
