| L(s) = 1 | + 2.63·3-s − 1.20·5-s + 4.34·7-s + 3.96·9-s + 5.31·11-s − 3.18·15-s − 0.324·17-s + 6.53·19-s + 11.4·21-s + 2.73·23-s − 3.53·25-s + 2.53·27-s − 8.00·29-s − 3.90·31-s + 14.0·33-s − 5.24·35-s − 4.32·37-s − 0.618·41-s − 9.93·43-s − 4.78·45-s − 1.31·47-s + 11.8·49-s − 0.855·51-s + 7.35·53-s − 6.41·55-s + 17.2·57-s + 4.69·59-s + ⋯ |
| L(s) = 1 | + 1.52·3-s − 0.540·5-s + 1.64·7-s + 1.32·9-s + 1.60·11-s − 0.823·15-s − 0.0786·17-s + 1.49·19-s + 2.50·21-s + 0.571·23-s − 0.707·25-s + 0.487·27-s − 1.48·29-s − 0.702·31-s + 2.43·33-s − 0.887·35-s − 0.711·37-s − 0.0965·41-s − 1.51·43-s − 0.713·45-s − 0.191·47-s + 1.69·49-s − 0.119·51-s + 1.01·53-s − 0.865·55-s + 2.28·57-s + 0.610·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.890989265\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.890989265\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 2.63T + 3T^{2} \) |
| 5 | \( 1 + 1.20T + 5T^{2} \) |
| 7 | \( 1 - 4.34T + 7T^{2} \) |
| 11 | \( 1 - 5.31T + 11T^{2} \) |
| 17 | \( 1 + 0.324T + 17T^{2} \) |
| 19 | \( 1 - 6.53T + 19T^{2} \) |
| 23 | \( 1 - 2.73T + 23T^{2} \) |
| 29 | \( 1 + 8.00T + 29T^{2} \) |
| 31 | \( 1 + 3.90T + 31T^{2} \) |
| 37 | \( 1 + 4.32T + 37T^{2} \) |
| 41 | \( 1 + 0.618T + 41T^{2} \) |
| 43 | \( 1 + 9.93T + 43T^{2} \) |
| 47 | \( 1 + 1.31T + 47T^{2} \) |
| 53 | \( 1 - 7.35T + 53T^{2} \) |
| 59 | \( 1 - 4.69T + 59T^{2} \) |
| 61 | \( 1 + 9.47T + 61T^{2} \) |
| 67 | \( 1 + 4.97T + 67T^{2} \) |
| 71 | \( 1 + 8.50T + 71T^{2} \) |
| 73 | \( 1 + 6.46T + 73T^{2} \) |
| 79 | \( 1 - 6.87T + 79T^{2} \) |
| 83 | \( 1 - 14.2T + 83T^{2} \) |
| 89 | \( 1 - 7.24T + 89T^{2} \) |
| 97 | \( 1 - 4.33T + 97T^{2} \) |
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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
−8.987139061822296097626173657898, −8.034520960853150877465039611257, −7.56939365888372599702139575277, −6.97796489956047893629440328663, −5.60374218308653505954485567656, −4.69516076824238755860190389301, −3.81912016351119121815768348808, −3.35061569397517417434388577036, −1.96865842596179731051986723217, −1.37991969792966475136613558095,
1.37991969792966475136613558095, 1.96865842596179731051986723217, 3.35061569397517417434388577036, 3.81912016351119121815768348808, 4.69516076824238755860190389301, 5.60374218308653505954485567656, 6.97796489956047893629440328663, 7.56939365888372599702139575277, 8.034520960853150877465039611257, 8.987139061822296097626173657898
