| L(s) = 1 | + 0.662·3-s − 9.39·5-s − 19.7·7-s − 26.5·9-s − 85.6·13-s − 6.22·15-s + 52.9·17-s − 108.·19-s − 13.0·21-s − 47.7·23-s − 36.7·25-s − 35.4·27-s + 289.·29-s + 11.3·31-s + 185.·35-s + 20.3·37-s − 56.7·39-s − 49.4·41-s − 441.·43-s + 249.·45-s + 273.·47-s + 46.1·49-s + 35.0·51-s − 92.8·53-s − 71.8·57-s + 670.·59-s + 141.·61-s + ⋯ |
| L(s) = 1 | + 0.127·3-s − 0.840·5-s − 1.06·7-s − 0.983·9-s − 1.82·13-s − 0.107·15-s + 0.755·17-s − 1.30·19-s − 0.135·21-s − 0.432·23-s − 0.294·25-s − 0.253·27-s + 1.85·29-s + 0.0660·31-s + 0.894·35-s + 0.0904·37-s − 0.233·39-s − 0.188·41-s − 1.56·43-s + 0.826·45-s + 0.850·47-s + 0.134·49-s + 0.0963·51-s − 0.240·53-s − 0.166·57-s + 1.47·59-s + 0.297·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4770863138\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4770863138\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 0.662T + 27T^{2} \) |
| 5 | \( 1 + 9.39T + 125T^{2} \) |
| 7 | \( 1 + 19.7T + 343T^{2} \) |
| 13 | \( 1 + 85.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 52.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 108.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 47.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 289.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 11.3T + 2.97e4T^{2} \) |
| 37 | \( 1 - 20.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 49.4T + 6.89e4T^{2} \) |
| 43 | \( 1 + 441.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 273.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 92.8T + 1.48e5T^{2} \) |
| 59 | \( 1 - 670.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 141.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 791.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 255.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 513.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.17e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 656.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 609.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 664.T + 9.12e5T^{2} \) |
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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
−9.830913878354492236121975039344, −8.647754838704591912765507781050, −8.060399981030380836280117133621, −7.09624927486472971141135253636, −6.32289265609023573785740797214, −5.25204915335047811369274435647, −4.22169406623817112411778632067, −3.18257423479738107920405189727, −2.39904947228753099657644494822, −0.34108306525941665111022515255,
0.34108306525941665111022515255, 2.39904947228753099657644494822, 3.18257423479738107920405189727, 4.22169406623817112411778632067, 5.25204915335047811369274435647, 6.32289265609023573785740797214, 7.09624927486472971141135253636, 8.060399981030380836280117133621, 8.647754838704591912765507781050, 9.830913878354492236121975039344
