| L(s) = 1 | + 1.63·3-s + 7-s − 0.328·9-s + 1.24·11-s + 4.20·13-s − 3.39·17-s + 6.46·19-s + 1.63·21-s − 2.15·23-s − 5.44·27-s + 3.96·29-s − 10.0·31-s + 2.02·33-s + 6.76·37-s + 6.87·39-s − 0.131·41-s + 7.40·43-s + 4.82·47-s + 49-s − 5.55·51-s + 10.0·53-s + 10.5·57-s + 10.9·59-s − 6.33·61-s − 0.328·63-s + 2.65·67-s − 3.52·69-s + ⋯ |
| L(s) = 1 | + 0.943·3-s + 0.377·7-s − 0.109·9-s + 0.374·11-s + 1.16·13-s − 0.823·17-s + 1.48·19-s + 0.356·21-s − 0.449·23-s − 1.04·27-s + 0.736·29-s − 1.80·31-s + 0.353·33-s + 1.11·37-s + 1.10·39-s − 0.0205·41-s + 1.12·43-s + 0.704·47-s + 0.142·49-s − 0.777·51-s + 1.37·53-s + 1.39·57-s + 1.43·59-s − 0.810·61-s − 0.0413·63-s + 0.324·67-s − 0.424·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.150230722\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.150230722\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 - 1.63T + 3T^{2} \) |
| 11 | \( 1 - 1.24T + 11T^{2} \) |
| 13 | \( 1 - 4.20T + 13T^{2} \) |
| 17 | \( 1 + 3.39T + 17T^{2} \) |
| 19 | \( 1 - 6.46T + 19T^{2} \) |
| 23 | \( 1 + 2.15T + 23T^{2} \) |
| 29 | \( 1 - 3.96T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 - 6.76T + 37T^{2} \) |
| 41 | \( 1 + 0.131T + 41T^{2} \) |
| 43 | \( 1 - 7.40T + 43T^{2} \) |
| 47 | \( 1 - 4.82T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 + 6.33T + 61T^{2} \) |
| 67 | \( 1 - 2.65T + 67T^{2} \) |
| 71 | \( 1 - 0.754T + 71T^{2} \) |
| 73 | \( 1 - 6.03T + 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 + 14.0T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 - 0.914T + 97T^{2} \) |
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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
−8.191386045497292633815809563333, −7.58217438278400769310095208448, −6.86496212838571758357652192570, −5.90441168255775178100858135691, −5.37602784852848891251523733766, −4.18094865025697697600304200708, −3.70335184009392336254936369134, −2.80777381438631753931589606134, −1.99410186194905043811262837703, −0.938037903893968933051656554678,
0.938037903893968933051656554678, 1.99410186194905043811262837703, 2.80777381438631753931589606134, 3.70335184009392336254936369134, 4.18094865025697697600304200708, 5.37602784852848891251523733766, 5.90441168255775178100858135691, 6.86496212838571758357652192570, 7.58217438278400769310095208448, 8.191386045497292633815809563333
