| L(s) = 1 | + 1.65·2-s − 2.37·3-s + 0.726·4-s − 3.92·6-s + 0.377·7-s − 2.10·8-s + 2.65·9-s − 1.37·11-s − 1.72·12-s − 2.82·13-s + 0.622·14-s − 4.92·16-s − 6.37·17-s + 4.37·18-s + 19-s − 0.896·21-s − 2.27·22-s − 6.19·23-s + 4.99·24-s − 4.65·26-s + 0.829·27-s + 0.273·28-s − 3.37·29-s + 2.48·31-s − 3.92·32-s + 3.27·33-s − 10.5·34-s + ⋯ |
| L(s) = 1 | + 1.16·2-s − 1.37·3-s + 0.363·4-s − 1.60·6-s + 0.142·7-s − 0.743·8-s + 0.883·9-s − 0.415·11-s − 0.498·12-s − 0.782·13-s + 0.166·14-s − 1.23·16-s − 1.54·17-s + 1.03·18-s + 0.229·19-s − 0.195·21-s − 0.484·22-s − 1.29·23-s + 1.02·24-s − 0.913·26-s + 0.159·27-s + 0.0517·28-s − 0.627·29-s + 0.445·31-s − 0.693·32-s + 0.569·33-s − 1.80·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 1.65T + 2T^{2} \) |
| 3 | \( 1 + 2.37T + 3T^{2} \) |
| 7 | \( 1 - 0.377T + 7T^{2} \) |
| 11 | \( 1 + 1.37T + 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 + 6.37T + 17T^{2} \) |
| 23 | \( 1 + 6.19T + 23T^{2} \) |
| 29 | \( 1 + 3.37T + 29T^{2} \) |
| 31 | \( 1 - 2.48T + 31T^{2} \) |
| 37 | \( 1 + 5.58T + 37T^{2} \) |
| 41 | \( 1 - 8.50T + 41T^{2} \) |
| 43 | \( 1 - 12.1T + 43T^{2} \) |
| 47 | \( 1 - 6.87T + 47T^{2} \) |
| 53 | \( 1 + 11.5T + 53T^{2} \) |
| 59 | \( 1 - 6.05T + 59T^{2} \) |
| 61 | \( 1 - 5.02T + 61T^{2} \) |
| 67 | \( 1 + 3.22T + 67T^{2} \) |
| 71 | \( 1 + 2.30T + 71T^{2} \) |
| 73 | \( 1 - 3.19T + 73T^{2} \) |
| 79 | \( 1 + 6.71T + 79T^{2} \) |
| 83 | \( 1 + 18.2T + 83T^{2} \) |
| 89 | \( 1 - 1.50T + 89T^{2} \) |
| 97 | \( 1 - 11.7T + 97T^{2} \) |
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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
−10.99557923397817890835730887741, −9.929999269860691039043409127078, −8.840502704715681938073667789669, −7.43647634996632542907942242136, −6.37029850032284263600142052056, −5.70185668950312104610222386817, −4.83824677001158109884965668823, −4.12650126542128489161939162915, −2.47046698591722202765913608213, 0,
2.47046698591722202765913608213, 4.12650126542128489161939162915, 4.83824677001158109884965668823, 5.70185668950312104610222386817, 6.37029850032284263600142052056, 7.43647634996632542907942242136, 8.840502704715681938073667789669, 9.929999269860691039043409127078, 10.99557923397817890835730887741
