| L(s) = 1 | + 3-s − 5-s + 3.60·7-s + 9-s + 1.60·11-s + 0.640·13-s − 15-s + 5.28·17-s − 19-s + 3.60·21-s + 8.24·23-s + 25-s + 27-s − 4.88·29-s + 7.28·31-s + 1.60·33-s − 3.60·35-s − 9.92·37-s + 0.640·39-s + 4.57·41-s − 7.92·43-s − 45-s − 0.249·47-s + 6.03·49-s + 5.28·51-s + 1.75·53-s − 1.60·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.36·7-s + 0.333·9-s + 0.485·11-s + 0.177·13-s − 0.258·15-s + 1.28·17-s − 0.229·19-s + 0.787·21-s + 1.72·23-s + 0.200·25-s + 0.192·27-s − 0.908·29-s + 1.30·31-s + 0.280·33-s − 0.610·35-s − 1.63·37-s + 0.102·39-s + 0.715·41-s − 1.20·43-s − 0.149·45-s − 0.0364·47-s + 0.861·49-s + 0.739·51-s + 0.240·53-s − 0.217·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.052524338\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.052524338\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 - 3.60T + 7T^{2} \) |
| 11 | \( 1 - 1.60T + 11T^{2} \) |
| 13 | \( 1 - 0.640T + 13T^{2} \) |
| 17 | \( 1 - 5.28T + 17T^{2} \) |
| 23 | \( 1 - 8.24T + 23T^{2} \) |
| 29 | \( 1 + 4.88T + 29T^{2} \) |
| 31 | \( 1 - 7.28T + 31T^{2} \) |
| 37 | \( 1 + 9.92T + 37T^{2} \) |
| 41 | \( 1 - 4.57T + 41T^{2} \) |
| 43 | \( 1 + 7.92T + 43T^{2} \) |
| 47 | \( 1 + 0.249T + 47T^{2} \) |
| 53 | \( 1 - 1.75T + 53T^{2} \) |
| 59 | \( 1 + 12.3T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 - 3.93T + 73T^{2} \) |
| 79 | \( 1 - 3.03T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 - 15.9T + 89T^{2} \) |
| 97 | \( 1 + 9.92T + 97T^{2} \) |
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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
−8.397749545896064533621763538596, −7.60800863840146645619356806621, −7.18911565528530066651119914657, −6.17474574443810042462218917804, −5.13213202654657089531764929083, −4.67686511391186482739441751976, −3.69006393834102253877941099068, −3.02793993035403799028998483863, −1.81029201872547652062376562438, −1.04619841104293910151690863677,
1.04619841104293910151690863677, 1.81029201872547652062376562438, 3.02793993035403799028998483863, 3.69006393834102253877941099068, 4.67686511391186482739441751976, 5.13213202654657089531764929083, 6.17474574443810042462218917804, 7.18911565528530066651119914657, 7.60800863840146645619356806621, 8.397749545896064533621763538596
