L(s) = 1 | + 2·4-s + (0.5 + 2.59i)7-s − 5.19i·13-s + 4·16-s + 5.19i·19-s − 5·25-s + (1 + 5.19i)28-s − 10.3i·31-s − 11·37-s − 8·43-s + (−6.5 + 2.59i)49-s − 10.3i·52-s + 15.5i·61-s + 8·64-s + 5·67-s + ⋯ |
L(s) = 1 | + 4-s + (0.188 + 0.981i)7-s − 1.44i·13-s + 16-s + 1.19i·19-s − 25-s + (0.188 + 0.981i)28-s − 1.86i·31-s − 1.80·37-s − 1.21·43-s + (−0.928 + 0.371i)49-s − 1.44i·52-s + 1.99i·61-s + 64-s + 0.610·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.188i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 - 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.42294 + 0.135678i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.42294 + 0.135678i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-0.5 - 2.59i)T \) |
good | 2 | \( 1 - 2T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 5.19iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 5.19iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 10.3iT - 31T^{2} \) |
| 37 | \( 1 + 11T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 15.5iT - 61T^{2} \) |
| 67 | \( 1 - 5T + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 15.5iT - 73T^{2} \) |
| 79 | \( 1 - 17T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 5.19iT - 97T^{2} \) |
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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
−12.31743923970937630724778633707, −11.77926194512005936178483948718, −10.66527287713730006787258232932, −9.806373410543754724936667305845, −8.338971354451228504286981770129, −7.60763725919989967678800386198, −6.14132080213471396691959727237, −5.43995460113139782364315789736, −3.41334002214837305559225227638, −2.05946601437150731894798750794,
1.79667931384199699777336608332, 3.53797810840314301150019977587, 4.96364125632316960128323566378, 6.66585914897251330746172488364, 7.08463020998111621518733980564, 8.404676935173241946472510450721, 9.715222368130451539338716214644, 10.76104049195539061702592615885, 11.44993374274485632181077985580, 12.34724376774533062031826258018