L(s) = 1 | + (−1.01 + 1.01i)2-s + (1.22 + 1.22i)3-s + 1.94i·4-s + (3.97 + 3.03i)5-s − 2.48·6-s + (1.87 − 1.87i)7-s + (−6.02 − 6.02i)8-s + 2.99i·9-s + (−7.09 + 0.953i)10-s − 10.7·11-s + (−2.38 + 2.38i)12-s + (15.5 + 15.5i)13-s + 3.78i·14-s + (1.15 + 8.58i)15-s + 4.41·16-s + (−13.8 + 13.8i)17-s + ⋯ |
L(s) = 1 | + (−0.506 + 0.506i)2-s + (0.408 + 0.408i)3-s + 0.487i·4-s + (0.794 + 0.606i)5-s − 0.413·6-s + (0.267 − 0.267i)7-s + (−0.753 − 0.753i)8-s + 0.333i·9-s + (−0.709 + 0.0953i)10-s − 0.973·11-s + (−0.198 + 0.198i)12-s + (1.19 + 1.19i)13-s + 0.270i·14-s + (0.0768 + 0.572i)15-s + 0.275·16-s + (−0.816 + 0.816i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.357 - 0.934i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.357 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.717042 + 1.04192i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.717042 + 1.04192i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.22 - 1.22i)T \) |
| 5 | \( 1 + (-3.97 - 3.03i)T \) |
| 7 | \( 1 + (-1.87 + 1.87i)T \) |
good | 2 | \( 1 + (1.01 - 1.01i)T - 4iT^{2} \) |
| 11 | \( 1 + 10.7T + 121T^{2} \) |
| 13 | \( 1 + (-15.5 - 15.5i)T + 169iT^{2} \) |
| 17 | \( 1 + (13.8 - 13.8i)T - 289iT^{2} \) |
| 19 | \( 1 + 33.1iT - 361T^{2} \) |
| 23 | \( 1 + (-8.39 - 8.39i)T + 529iT^{2} \) |
| 29 | \( 1 + 42.0iT - 841T^{2} \) |
| 31 | \( 1 - 44.7T + 961T^{2} \) |
| 37 | \( 1 + (-21.6 + 21.6i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 22.6T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-9.41 - 9.41i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-46.0 + 46.0i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-51.4 - 51.4i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 - 5.55iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 49.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + (38.8 - 38.8i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 23.5T + 5.04e3T^{2} \) |
| 73 | \( 1 + (60.1 + 60.1i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 - 8.04iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (5.76 + 5.76i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + 62.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-21.9 + 21.9i)T - 9.40e3iT^{2} \) |
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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
−13.52469108394550638072046519305, −13.42173353103811602386977724475, −11.47712156376904132378847558992, −10.52674302939643162525610836752, −9.286709877777656911093724668842, −8.516892327175528067339633706451, −7.22572080491040419019518166261, −6.18774010941217512457482886476, −4.25376200906677366995727186218, −2.60175289541509538553588069582,
1.20092475946463513272744461365, 2.68791798712816473359462782124, 5.19459845650228502326731994476, 6.16213448371115045493357819958, 8.141539380765861750442813767814, 8.831263887569637731790553049913, 10.04868425684991464342398230004, 10.80087061230646201005671712444, 12.21147022705578314173999537839, 13.21444429569522309338613465994