| L(s) = 1 | + 3·3-s + 5.11·5-s + 7·7-s + 9·9-s − 15.1·11-s − 26·13-s + 15.3·15-s − 11.3·17-s + 10.2·19-s + 21·21-s − 166.·23-s − 98.8·25-s + 27·27-s − 36.2·29-s − 16.8·31-s − 45.3·33-s + 35.8·35-s − 295.·37-s − 78·39-s + 256.·41-s − 137.·43-s + 46.0·45-s − 93.3·47-s + 49·49-s − 34.0·51-s − 391.·53-s − 77.3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.457·5-s + 0.377·7-s + 0.333·9-s − 0.414·11-s − 0.554·13-s + 0.264·15-s − 0.161·17-s + 0.123·19-s + 0.218·21-s − 1.50·23-s − 0.790·25-s + 0.192·27-s − 0.231·29-s − 0.0976·31-s − 0.239·33-s + 0.172·35-s − 1.31·37-s − 0.320·39-s + 0.976·41-s − 0.488·43-s + 0.152·45-s − 0.289·47-s + 0.142·49-s − 0.0934·51-s − 1.01·53-s − 0.189·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 - 7T \) |
| good | 5 | \( 1 - 5.11T + 125T^{2} \) |
| 11 | \( 1 + 15.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 26T + 2.19e3T^{2} \) |
| 17 | \( 1 + 11.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 10.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 166.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 36.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 16.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 295.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 256.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 137.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 93.3T + 1.03e5T^{2} \) |
| 53 | \( 1 + 391.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 213.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 546.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 58.5T + 3.00e5T^{2} \) |
| 71 | \( 1 + 231.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 714.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 93.3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 979.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 358.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 187.T + 9.12e5T^{2} \) |
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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
−8.833411618260245359346916238143, −7.987398725497836190745291694159, −7.42485115674953183799330804750, −6.33740003968903892919029801825, −5.46917015801311253526488378295, −4.54578020812063873749130741946, −3.55354672843951218922981847574, −2.40152105172818276218621504678, −1.65860971437973986625258510367, 0,
1.65860971437973986625258510367, 2.40152105172818276218621504678, 3.55354672843951218922981847574, 4.54578020812063873749130741946, 5.46917015801311253526488378295, 6.33740003968903892919029801825, 7.42485115674953183799330804750, 7.987398725497836190745291694159, 8.833411618260245359346916238143
