L(s) = 1 | + (−1.61 + 0.618i)3-s + 3.23·5-s + 1.23i·7-s + (2.23 − 2.00i)9-s − 5.23i·11-s − 4.47i·13-s + (−5.23 + 2.00i)15-s + 2.47i·17-s + 0.763·19-s + (−0.763 − 2.00i)21-s + 2.47·23-s + 5.47·25-s + (−2.38 + 4.61i)27-s − 4.76·29-s − 5.23i·31-s + ⋯ |
L(s) = 1 | + (−0.934 + 0.356i)3-s + 1.44·5-s + 0.467i·7-s + (0.745 − 0.666i)9-s − 1.57i·11-s − 1.24i·13-s + (−1.35 + 0.516i)15-s + 0.599i·17-s + 0.175·19-s + (−0.166 − 0.436i)21-s + 0.515·23-s + 1.09·25-s + (−0.458 + 0.888i)27-s − 0.884·29-s − 0.940i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 + 0.408i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.912 + 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.41754 - 0.302535i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.41754 - 0.302535i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.61 - 0.618i)T \) |
good | 5 | \( 1 - 3.23T + 5T^{2} \) |
| 7 | \( 1 - 1.23iT - 7T^{2} \) |
| 11 | \( 1 + 5.23iT - 11T^{2} \) |
| 13 | \( 1 + 4.47iT - 13T^{2} \) |
| 17 | \( 1 - 2.47iT - 17T^{2} \) |
| 19 | \( 1 - 0.763T + 19T^{2} \) |
| 23 | \( 1 - 2.47T + 23T^{2} \) |
| 29 | \( 1 + 4.76T + 29T^{2} \) |
| 31 | \( 1 + 5.23iT - 31T^{2} \) |
| 37 | \( 1 + 8.47iT - 37T^{2} \) |
| 41 | \( 1 - 6.47iT - 41T^{2} \) |
| 43 | \( 1 - 7.23T + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 - 3.23T + 53T^{2} \) |
| 59 | \( 1 - 1.23iT - 59T^{2} \) |
| 61 | \( 1 - 0.472iT - 61T^{2} \) |
| 67 | \( 1 - 9.70T + 67T^{2} \) |
| 71 | \( 1 + 15.4T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + 0.291iT - 79T^{2} \) |
| 83 | \( 1 - 2.76iT - 83T^{2} \) |
| 89 | \( 1 - 4iT - 89T^{2} \) |
| 97 | \( 1 - 0.472T + 97T^{2} \) |
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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
−10.47616315024348821982939457796, −9.455314854616096544133205954184, −8.866509056968612168904156083107, −7.63923977120824812992980645014, −6.27909751126357004330830435575, −5.70312602719558261373858841543, −5.40736040874143492967607809426, −3.81326120014705840383825473100, −2.54211825191188834926676242480, −0.925651741610071157297040918962,
1.43455523546007024285684201262, 2.29049025474332987966901383720, 4.30563125799170975452305091893, 5.08875230228693152536572343495, 5.97262787248073421141463675430, 7.05093251566331885525166915368, 7.19509138506185627797900867530, 8.966734072996930601979750704397, 9.728372109490548707837226539293, 10.27106252088859004387537310155