L(s) = 1 | + (0.766 − 0.642i)3-s + (3.41 + 1.24i)5-s + (−2.06 − 0.752i)7-s + (0.173 − 0.984i)9-s + (−2.44 + 4.23i)11-s + (1.12 + 6.35i)13-s + (3.41 − 1.24i)15-s + (−1.15 + 6.55i)17-s + (6.22 − 5.22i)19-s + (−2.06 + 0.752i)21-s + (1.09 + 1.90i)23-s + (6.27 + 5.26i)25-s + (−0.500 − 0.866i)27-s + (3.05 − 5.29i)29-s − 1.06·31-s + ⋯ |
L(s) = 1 | + (0.442 − 0.371i)3-s + (1.52 + 0.555i)5-s + (−0.781 − 0.284i)7-s + (0.0578 − 0.328i)9-s + (−0.736 + 1.27i)11-s + (0.310 + 1.76i)13-s + (0.881 − 0.320i)15-s + (−0.280 + 1.58i)17-s + (1.42 − 1.19i)19-s + (−0.451 + 0.164i)21-s + (0.229 + 0.397i)23-s + (1.25 + 1.05i)25-s + (−0.0962 − 0.166i)27-s + (0.567 − 0.982i)29-s − 0.191·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 - 0.592i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.01917 + 0.662871i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.01917 + 0.662871i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.766 + 0.642i)T \) |
| 37 | \( 1 + (0.272 + 6.07i)T \) |
good | 5 | \( 1 + (-3.41 - 1.24i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (2.06 + 0.752i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (2.44 - 4.23i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.12 - 6.35i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (1.15 - 6.55i)T + (-15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (-6.22 + 5.22i)T + (3.29 - 18.7i)T^{2} \) |
| 23 | \( 1 + (-1.09 - 1.90i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.05 + 5.29i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 1.06T + 31T^{2} \) |
| 41 | \( 1 + (0.391 + 2.21i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + 2.91T + 43T^{2} \) |
| 47 | \( 1 + (1.92 + 3.34i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.976 + 0.355i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-9.96 + 3.62i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (1.27 + 7.22i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (2.25 + 0.822i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-0.202 + 0.169i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + 3.97T + 73T^{2} \) |
| 79 | \( 1 + (1.63 + 0.595i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (0.645 - 3.66i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (-12.1 + 4.40i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-4.45 - 7.72i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.889926027702811708815435812347, −9.601639466900312054918811579044, −8.764533498454182078148367209100, −7.38234737338064862930852234236, −6.76877082108560141548270701996, −6.15624851638853069091361528542, −4.97921150906213996483223353610, −3.70653263146332391904460114161, −2.42310497270677450465047135958, −1.76097704305436014179821258206,
1.02124441916503383381022153786, 2.86248445779061680038404475077, 3.13183224549520465511665949097, 5.14833274322065370470732510576, 5.48081341711199954874053557735, 6.32684022878006127748982466039, 7.69233780196789199793272902982, 8.581124706095340747830138237977, 9.245213099781676515134910756598, 10.09884612953323520761291773915