| L(s) = 1 | − 0.295·3-s − 3.76·5-s + 1.37·7-s − 2.91·9-s + 4.17·11-s − 2.50·13-s + 1.11·15-s − 5.93·17-s + 0.213·19-s − 0.405·21-s + 6.49·23-s + 9.15·25-s + 1.74·27-s + 6.78·29-s − 6.95·31-s − 1.23·33-s − 5.15·35-s + 6.45·37-s + 0.739·39-s + 3.06·41-s − 3.86·43-s + 10.9·45-s + 4.00·47-s − 5.12·49-s + 1.75·51-s + 12.1·53-s − 15.7·55-s + ⋯ |
| L(s) = 1 | − 0.170·3-s − 1.68·5-s + 0.518·7-s − 0.970·9-s + 1.25·11-s − 0.694·13-s + 0.287·15-s − 1.43·17-s + 0.0490·19-s − 0.0883·21-s + 1.35·23-s + 1.83·25-s + 0.336·27-s + 1.25·29-s − 1.24·31-s − 0.214·33-s − 0.871·35-s + 1.06·37-s + 0.118·39-s + 0.477·41-s − 0.588·43-s + 1.63·45-s + 0.584·47-s − 0.731·49-s + 0.245·51-s + 1.66·53-s − 2.11·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9791048890\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9791048890\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 167 | \( 1 + T \) |
| good | 3 | \( 1 + 0.295T + 3T^{2} \) |
| 5 | \( 1 + 3.76T + 5T^{2} \) |
| 7 | \( 1 - 1.37T + 7T^{2} \) |
| 11 | \( 1 - 4.17T + 11T^{2} \) |
| 13 | \( 1 + 2.50T + 13T^{2} \) |
| 17 | \( 1 + 5.93T + 17T^{2} \) |
| 19 | \( 1 - 0.213T + 19T^{2} \) |
| 23 | \( 1 - 6.49T + 23T^{2} \) |
| 29 | \( 1 - 6.78T + 29T^{2} \) |
| 31 | \( 1 + 6.95T + 31T^{2} \) |
| 37 | \( 1 - 6.45T + 37T^{2} \) |
| 41 | \( 1 - 3.06T + 41T^{2} \) |
| 43 | \( 1 + 3.86T + 43T^{2} \) |
| 47 | \( 1 - 4.00T + 47T^{2} \) |
| 53 | \( 1 - 12.1T + 53T^{2} \) |
| 59 | \( 1 - 13.3T + 59T^{2} \) |
| 61 | \( 1 - 1.28T + 61T^{2} \) |
| 67 | \( 1 + 2.37T + 67T^{2} \) |
| 71 | \( 1 - 8.31T + 71T^{2} \) |
| 73 | \( 1 - 2.88T + 73T^{2} \) |
| 79 | \( 1 - 13.8T + 79T^{2} \) |
| 83 | \( 1 - 8.20T + 83T^{2} \) |
| 89 | \( 1 + 1.96T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−9.350512800845932799220242446959, −8.701390055269997149263174185643, −8.109275003859353983510372253450, −7.11561537771986121361726079153, −6.59792427772702446724055417829, −5.21716421884482142048154964209, −4.43720136967558529043971803071, −3.67362843340682248687236920766, −2.52787812721698619186031186493, −0.71760205230954866711681060204,
0.71760205230954866711681060204, 2.52787812721698619186031186493, 3.67362843340682248687236920766, 4.43720136967558529043971803071, 5.21716421884482142048154964209, 6.59792427772702446724055417829, 7.11561537771986121361726079153, 8.109275003859353983510372253450, 8.701390055269997149263174185643, 9.350512800845932799220242446959
