| L(s) = 1 | + 1.30·3-s + 1.30·5-s − 2.30·7-s − 1.30·9-s + 11-s − 1.69·13-s + 1.69·15-s − 2.60·17-s + 19-s − 3·21-s + 4.60·23-s − 3.30·25-s − 5.60·27-s − 9.30·29-s − 2.90·31-s + 1.30·33-s − 3·35-s + 8·37-s − 2.21·39-s − 3.69·41-s − 1.30·43-s − 1.69·45-s − 3.21·47-s − 1.69·49-s − 3.39·51-s − 0.605·53-s + 1.30·55-s + ⋯ |
| L(s) = 1 | + 0.752·3-s + 0.582·5-s − 0.870·7-s − 0.434·9-s + 0.301·11-s − 0.470·13-s + 0.438·15-s − 0.631·17-s + 0.229·19-s − 0.654·21-s + 0.960·23-s − 0.660·25-s − 1.07·27-s − 1.72·29-s − 0.522·31-s + 0.226·33-s − 0.507·35-s + 1.31·37-s − 0.354·39-s − 0.577·41-s − 0.198·43-s − 0.253·45-s − 0.468·47-s − 0.242·49-s − 0.475·51-s − 0.0831·53-s + 0.175·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 1.30T + 3T^{2} \) |
| 5 | \( 1 - 1.30T + 5T^{2} \) |
| 7 | \( 1 + 2.30T + 7T^{2} \) |
| 13 | \( 1 + 1.69T + 13T^{2} \) |
| 17 | \( 1 + 2.60T + 17T^{2} \) |
| 23 | \( 1 - 4.60T + 23T^{2} \) |
| 29 | \( 1 + 9.30T + 29T^{2} \) |
| 31 | \( 1 + 2.90T + 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 + 3.69T + 41T^{2} \) |
| 43 | \( 1 + 1.30T + 43T^{2} \) |
| 47 | \( 1 + 3.21T + 47T^{2} \) |
| 53 | \( 1 + 0.605T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 5.51T + 67T^{2} \) |
| 71 | \( 1 - 2.09T + 71T^{2} \) |
| 73 | \( 1 + 16.6T + 73T^{2} \) |
| 79 | \( 1 + 9.81T + 79T^{2} \) |
| 83 | \( 1 + 0.0916T + 83T^{2} \) |
| 89 | \( 1 + 7.81T + 89T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
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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
−8.358895437090235206759436796860, −7.50131396194742971695020905811, −6.81706128950577403993615636276, −5.96821453965534371336900433131, −5.35753755528923034961581183438, −4.19418652498217995583052543980, −3.33235690774399847798854618134, −2.62687982149335497551463432781, −1.72584894135688654187460871474, 0,
1.72584894135688654187460871474, 2.62687982149335497551463432781, 3.33235690774399847798854618134, 4.19418652498217995583052543980, 5.35753755528923034961581183438, 5.96821453965534371336900433131, 6.81706128950577403993615636276, 7.50131396194742971695020905811, 8.358895437090235206759436796860
