| L(s) = 1 | − 4i·2-s + (−14.2 − 6.28i)3-s − 16·4-s + 46.4·5-s + (−25.1 + 57.0i)6-s − 104. i·7-s + 64i·8-s + (164. + 179. i)9-s − 185. i·10-s − 578.·11-s + (228. + 100. i)12-s − 661.·13-s − 419.·14-s + (−662. − 291. i)15-s + 256·16-s − 54.6·17-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + (−0.915 − 0.403i)3-s − 0.5·4-s + 0.830·5-s + (−0.285 + 0.647i)6-s − 0.809i·7-s + 0.353i·8-s + (0.674 + 0.737i)9-s − 0.587i·10-s − 1.44·11-s + (0.457 + 0.201i)12-s − 1.08·13-s − 0.572·14-s + (−0.759 − 0.334i)15-s + 0.250·16-s − 0.0458·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.686 - 0.726i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.686 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.5456159098\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5456159098\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4iT \) |
| 3 | \( 1 + (14.2 + 6.28i)T \) |
| 23 | \( 1 + (-2.39e3 - 851. i)T \) |
| good | 5 | \( 1 - 46.4T + 3.12e3T^{2} \) |
| 7 | \( 1 + 104. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 578.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 661.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 54.6T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.05e3iT - 2.47e6T^{2} \) |
| 29 | \( 1 - 2.90e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 8.28e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.25e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 7.39e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 3.07e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 2.20e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 2.46e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 8.21e3iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 5.50e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 6.61e3iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 2.09e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 - 4.87e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.46e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 6.29e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.14e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.17e4iT - 8.58e9T^{2} \) |
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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
−12.41590532708225949262871260651, −11.34493303647437150047395337004, −10.21692893851076933977849874292, −9.984462990798433315890623251103, −8.066821972399731497493089370722, −6.96217053070911173373745288950, −5.54866489353245985273224267953, −4.66984589001527914903431436633, −2.65100692864205695340297777601, −1.23622446768516260163824371217,
0.22685648466957355598248453245, 2.52511618355818229315172955085, 4.77672158712513882094488399468, 5.44718262599895643755734196803, 6.41675504445924504886076649342, 7.69499119490237621498785007646, 9.180015717810545066775766953364, 9.949660959892987772913427900803, 10.96972609479518483794088022240, 12.32092317412618838544636595681
