| L(s) = 1 | + 6.10·3-s − 29.3·7-s + 10.2·9-s − 14.8·11-s + 30.7·13-s + 9.12·17-s + 124.·19-s − 179.·21-s + 82.3·23-s − 102.·27-s − 32.9·29-s + 173.·31-s − 90.8·33-s − 83.2·37-s + 187.·39-s − 378.·41-s − 274.·43-s + 238.·47-s + 516.·49-s + 55.6·51-s + 41.2·53-s + 762.·57-s + 274.·59-s + 75.6·61-s − 301.·63-s − 1.06e3·67-s + 502.·69-s + ⋯ |
| L(s) = 1 | + 1.17·3-s − 1.58·7-s + 0.381·9-s − 0.407·11-s + 0.655·13-s + 0.130·17-s + 1.50·19-s − 1.86·21-s + 0.746·23-s − 0.727·27-s − 0.211·29-s + 1.00·31-s − 0.479·33-s − 0.369·37-s + 0.770·39-s − 1.44·41-s − 0.975·43-s + 0.741·47-s + 1.50·49-s + 0.152·51-s + 0.106·53-s + 1.77·57-s + 0.604·59-s + 0.158·61-s − 0.603·63-s − 1.93·67-s + 0.876·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2000 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 6.10T + 27T^{2} \) |
| 7 | \( 1 + 29.3T + 343T^{2} \) |
| 11 | \( 1 + 14.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 30.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 9.12T + 4.91e3T^{2} \) |
| 19 | \( 1 - 124.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 82.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 32.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 173.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 83.2T + 5.06e4T^{2} \) |
| 41 | \( 1 + 378.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 274.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 238.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 41.2T + 1.48e5T^{2} \) |
| 59 | \( 1 - 274.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 75.6T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.06e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 924.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 730.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 822.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 480.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.04e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 597.T + 9.12e5T^{2} \) |
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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
−8.641528488854101997815413251794, −7.63950162027163243907239390934, −6.99486280176465178771968208380, −6.10224772138829155549040181618, −5.25376951225060230571395691720, −3.93590694474672046835704125536, −3.09730970334352067756099409409, −2.85289374067237507069978539280, −1.36922781207900261396082013057, 0,
1.36922781207900261396082013057, 2.85289374067237507069978539280, 3.09730970334352067756099409409, 3.93590694474672046835704125536, 5.25376951225060230571395691720, 6.10224772138829155549040181618, 6.99486280176465178771968208380, 7.63950162027163243907239390934, 8.641528488854101997815413251794
