| L(s) = 1 | − 2.77·3-s + 1.76·5-s − 4.75·7-s + 4.69·9-s + 1.21·11-s + 1.19·13-s − 4.90·15-s − 17-s − 2.51·19-s + 13.1·21-s − 7.49·23-s − 1.87·25-s − 4.69·27-s + 5.63·29-s − 7.04·31-s − 3.37·33-s − 8.40·35-s + 8.00·37-s − 3.32·39-s − 8.01·41-s + 3.67·43-s + 8.29·45-s − 0.175·47-s + 15.6·49-s + 2.77·51-s − 5.21·53-s + 2.15·55-s + ⋯ |
| L(s) = 1 | − 1.60·3-s + 0.791·5-s − 1.79·7-s + 1.56·9-s + 0.367·11-s + 0.332·13-s − 1.26·15-s − 0.242·17-s − 0.577·19-s + 2.87·21-s − 1.56·23-s − 0.374·25-s − 0.903·27-s + 1.04·29-s − 1.26·31-s − 0.587·33-s − 1.42·35-s + 1.31·37-s − 0.532·39-s − 1.25·41-s + 0.560·43-s + 1.23·45-s − 0.0256·47-s + 2.22·49-s + 0.388·51-s − 0.715·53-s + 0.290·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4904081721\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4904081721\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 + 2.77T + 3T^{2} \) |
| 5 | \( 1 - 1.76T + 5T^{2} \) |
| 7 | \( 1 + 4.75T + 7T^{2} \) |
| 11 | \( 1 - 1.21T + 11T^{2} \) |
| 13 | \( 1 - 1.19T + 13T^{2} \) |
| 19 | \( 1 + 2.51T + 19T^{2} \) |
| 23 | \( 1 + 7.49T + 23T^{2} \) |
| 29 | \( 1 - 5.63T + 29T^{2} \) |
| 31 | \( 1 + 7.04T + 31T^{2} \) |
| 37 | \( 1 - 8.00T + 37T^{2} \) |
| 41 | \( 1 + 8.01T + 41T^{2} \) |
| 43 | \( 1 - 3.67T + 43T^{2} \) |
| 47 | \( 1 + 0.175T + 47T^{2} \) |
| 53 | \( 1 + 5.21T + 53T^{2} \) |
| 61 | \( 1 + 3.24T + 61T^{2} \) |
| 67 | \( 1 - 4.05T + 67T^{2} \) |
| 71 | \( 1 - 4.37T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 + 4.84T + 79T^{2} \) |
| 83 | \( 1 - 9.63T + 83T^{2} \) |
| 89 | \( 1 - 1.09T + 89T^{2} \) |
| 97 | \( 1 + 13.9T + 97T^{2} \) |
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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
−7.58013181607921175382873255007, −6.65493447106811599521265491289, −6.33731133800307345546772045396, −5.97694546308824150832625750329, −5.33274002334664757538887256333, −4.31699536985059741306218171908, −3.69840631352331424179932399142, −2.61221017449285361273956843354, −1.57983302386006586870935123851, −0.37584402748764985038978514128,
0.37584402748764985038978514128, 1.57983302386006586870935123851, 2.61221017449285361273956843354, 3.69840631352331424179932399142, 4.31699536985059741306218171908, 5.33274002334664757538887256333, 5.97694546308824150832625750329, 6.33731133800307345546772045396, 6.65493447106811599521265491289, 7.58013181607921175382873255007
