| L(s) = 1 | + 2.79·3-s + 2.30·7-s + 4.82·9-s − 5.01·11-s − 1.36·13-s − 7.56·17-s + 4.96·19-s + 6.44·21-s − 0.225·23-s + 5.09·27-s + 8.18·29-s + 6.87·31-s − 14.0·33-s + 6.20·37-s − 3.81·39-s + 4.69·41-s + 4.30·43-s + 4.39·47-s − 1.69·49-s − 21.1·51-s + 10.4·53-s + 13.8·57-s + 0.0364·59-s − 9.99·61-s + 11.1·63-s + 14.6·67-s − 0.629·69-s + ⋯ |
| L(s) = 1 | + 1.61·3-s + 0.870·7-s + 1.60·9-s − 1.51·11-s − 0.378·13-s − 1.83·17-s + 1.13·19-s + 1.40·21-s − 0.0469·23-s + 0.981·27-s + 1.52·29-s + 1.23·31-s − 2.44·33-s + 1.02·37-s − 0.610·39-s + 0.733·41-s + 0.655·43-s + 0.641·47-s − 0.242·49-s − 2.96·51-s + 1.43·53-s + 1.83·57-s + 0.00474·59-s − 1.27·61-s + 1.39·63-s + 1.79·67-s − 0.0758·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.981616360\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.981616360\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 2.79T + 3T^{2} \) |
| 7 | \( 1 - 2.30T + 7T^{2} \) |
| 11 | \( 1 + 5.01T + 11T^{2} \) |
| 13 | \( 1 + 1.36T + 13T^{2} \) |
| 17 | \( 1 + 7.56T + 17T^{2} \) |
| 19 | \( 1 - 4.96T + 19T^{2} \) |
| 23 | \( 1 + 0.225T + 23T^{2} \) |
| 29 | \( 1 - 8.18T + 29T^{2} \) |
| 31 | \( 1 - 6.87T + 31T^{2} \) |
| 37 | \( 1 - 6.20T + 37T^{2} \) |
| 41 | \( 1 - 4.69T + 41T^{2} \) |
| 43 | \( 1 - 4.30T + 43T^{2} \) |
| 47 | \( 1 - 4.39T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 - 0.0364T + 59T^{2} \) |
| 61 | \( 1 + 9.99T + 61T^{2} \) |
| 67 | \( 1 - 14.6T + 67T^{2} \) |
| 71 | \( 1 - 1.24T + 71T^{2} \) |
| 73 | \( 1 - 3.30T + 73T^{2} \) |
| 79 | \( 1 - 8.12T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 + 0.854T + 89T^{2} \) |
| 97 | \( 1 - 5.51T + 97T^{2} \) |
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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
−7.921017311165393302187675364785, −7.47191473400121007490794106499, −6.70812633601874012256091530376, −5.66093364338556333841122738752, −4.64927477363016494040244262219, −4.47139986313448252684904374418, −3.27492908719037847736024908309, −2.42500412737178918327211903778, −2.31940153245486958530738604257, −0.912360102410202237244780234330,
0.912360102410202237244780234330, 2.31940153245486958530738604257, 2.42500412737178918327211903778, 3.27492908719037847736024908309, 4.47139986313448252684904374418, 4.64927477363016494040244262219, 5.66093364338556333841122738752, 6.70812633601874012256091530376, 7.47191473400121007490794106499, 7.921017311165393302187675364785
