| L(s) = 1 | + 2-s − 0.732·3-s + 4-s − 0.732·6-s + 1.26·7-s + 8-s − 2.46·9-s − 0.732·12-s − 2.46·13-s + 1.26·14-s + 16-s + 1.73·17-s − 2.46·18-s + 4.19·19-s − 0.928·21-s − 2.73·23-s − 0.732·24-s − 2.46·26-s + 4·27-s + 1.26·28-s − 3.73·29-s + 8.73·31-s + 32-s + 1.73·34-s − 2.46·36-s + 7.92·37-s + 4.19·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.422·3-s + 0.5·4-s − 0.298·6-s + 0.479·7-s + 0.353·8-s − 0.821·9-s − 0.211·12-s − 0.683·13-s + 0.338·14-s + 0.250·16-s + 0.420·17-s − 0.580·18-s + 0.962·19-s − 0.202·21-s − 0.569·23-s − 0.149·24-s − 0.483·26-s + 0.769·27-s + 0.239·28-s − 0.693·29-s + 1.56·31-s + 0.176·32-s + 0.297·34-s − 0.410·36-s + 1.30·37-s + 0.680·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.592709153\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.592709153\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 0.732T + 3T^{2} \) |
| 7 | \( 1 - 1.26T + 7T^{2} \) |
| 13 | \( 1 + 2.46T + 13T^{2} \) |
| 17 | \( 1 - 1.73T + 17T^{2} \) |
| 19 | \( 1 - 4.19T + 19T^{2} \) |
| 23 | \( 1 + 2.73T + 23T^{2} \) |
| 29 | \( 1 + 3.73T + 29T^{2} \) |
| 31 | \( 1 - 8.73T + 31T^{2} \) |
| 37 | \( 1 - 7.92T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 - 9.46T + 43T^{2} \) |
| 47 | \( 1 + 6.73T + 47T^{2} \) |
| 53 | \( 1 + 3T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 + 8.92T + 61T^{2} \) |
| 67 | \( 1 + 7.26T + 67T^{2} \) |
| 71 | \( 1 - 6.92T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 - 7.26T + 79T^{2} \) |
| 83 | \( 1 + 3.26T + 83T^{2} \) |
| 89 | \( 1 - 0.464T + 89T^{2} \) |
| 97 | \( 1 - 17.1T + 97T^{2} \) |
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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
−7.914646407340847997648108467262, −7.38167966228625928123347989788, −6.38718174763970374989242801938, −5.92324225279773662234803813091, −5.05451339847135505803112642545, −4.74580881047091908250412799218, −3.62514230123337087270518329937, −2.88820883183210235604335550566, −2.01493014116983020359536624024, −0.77036272850290418904708364104,
0.77036272850290418904708364104, 2.01493014116983020359536624024, 2.88820883183210235604335550566, 3.62514230123337087270518329937, 4.74580881047091908250412799218, 5.05451339847135505803112642545, 5.92324225279773662234803813091, 6.38718174763970374989242801938, 7.38167966228625928123347989788, 7.914646407340847997648108467262
