L(s) = 1 | + (−0.707 + 1.22i)2-s + (−0.499 − 0.866i)4-s + (0.965 − 0.258i)5-s − i·7-s + (−0.366 + 1.36i)10-s − i·13-s + (1.22 + 0.707i)14-s + (0.499 − 0.866i)16-s + (0.707 + 1.22i)17-s + (−0.707 − 0.707i)20-s + (0.866 − 0.499i)25-s + (1.22 + 0.707i)26-s + (−0.866 + 0.499i)28-s − 1.41i·29-s + (0.5 + 0.866i)31-s + (0.707 + 1.22i)32-s + ⋯ |
L(s) = 1 | + (−0.707 + 1.22i)2-s + (−0.499 − 0.866i)4-s + (0.965 − 0.258i)5-s − i·7-s + (−0.366 + 1.36i)10-s − i·13-s + (1.22 + 0.707i)14-s + (0.499 − 0.866i)16-s + (0.707 + 1.22i)17-s + (−0.707 − 0.707i)20-s + (0.866 − 0.499i)25-s + (1.22 + 0.707i)26-s + (−0.866 + 0.499i)28-s − 1.41i·29-s + (0.5 + 0.866i)31-s + (0.707 + 1.22i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.660 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.660 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8124227057\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8124227057\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.965 + 0.258i)T \) |
| 7 | \( 1 + iT \) |
good | 2 | \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + iT - T^{2} \) |
| 17 | \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + 1.41iT - T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + 1.41iT - T^{2} \) |
| 43 | \( 1 - iT - T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - 1.41iT - T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + 1.41T + T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 - iT - T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.15190171676344183713981344935, −9.479637335753658067986597308173, −8.437930792390810824242491803169, −7.910867975832086134365975795373, −7.04833390330166839296362650763, −6.10261372951973453635744560912, −5.63461009599777935483770439415, −4.39882049913895339530161195320, −2.98273384987733211246036718508, −1.17485937455648998644563747817,
1.52953320984186407294471071750, 2.46966594963393965557838294889, 3.22722846851958959061324868924, 4.83089269199131066783555468776, 5.84182549058818322409302114300, 6.65482622010402470957291422329, 7.915882590085191585288852352483, 9.147318956831373307720984182003, 9.260418990238290683976919462257, 10.02821762261276181861832531286