L(s) = 1 | + 2.59i·2-s + (−1.34 + 1.09i)3-s − 4.72·4-s + (−0.626 − 1.08i)5-s + (−2.84 − 3.47i)6-s − 7.07i·8-s + (0.599 − 2.93i)9-s + (2.81 − 1.62i)10-s + (−0.534 − 0.308i)11-s + (6.34 − 5.17i)12-s + (−1.06 − 0.613i)13-s + (2.02 + 0.769i)15-s + 8.88·16-s + (−2.21 − 3.83i)17-s + (7.62 + 1.55i)18-s + (1.64 + 0.950i)19-s + ⋯ |
L(s) = 1 | + 1.83i·2-s + (−0.774 + 0.632i)3-s − 2.36·4-s + (−0.280 − 0.485i)5-s + (−1.16 − 1.42i)6-s − 2.50i·8-s + (0.199 − 0.979i)9-s + (0.889 − 0.513i)10-s + (−0.161 − 0.0929i)11-s + (1.83 − 1.49i)12-s + (−0.294 − 0.170i)13-s + (0.523 + 0.198i)15-s + 2.22·16-s + (−0.537 − 0.930i)17-s + (1.79 + 0.366i)18-s + (0.377 + 0.218i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0367i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0367i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.288229 + 0.00530331i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.288229 + 0.00530331i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.34 - 1.09i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 2.59iT - 2T^{2} \) |
| 5 | \( 1 + (0.626 + 1.08i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (0.534 + 0.308i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.06 + 0.613i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (2.21 + 3.83i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.64 - 0.950i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4.11 - 2.37i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (5.07 - 2.93i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 2.48iT - 31T^{2} \) |
| 37 | \( 1 + (-1.33 + 2.30i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.09 + 3.63i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.24 + 3.89i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 7.61T + 47T^{2} \) |
| 53 | \( 1 + (2.67 - 1.54i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 3.56T + 59T^{2} \) |
| 61 | \( 1 + 14.4iT - 61T^{2} \) |
| 67 | \( 1 - 13.6T + 67T^{2} \) |
| 71 | \( 1 + 10.4iT - 71T^{2} \) |
| 73 | \( 1 + (9.95 - 5.74i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + 4.03T + 79T^{2} \) |
| 83 | \( 1 + (-4.36 - 7.56i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (0.811 - 1.40i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (8.76 - 5.06i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.07021082047027102861510486255, −9.789120227983131521240027650153, −9.223752243357368211145164337674, −8.204670711209490087828197831774, −7.30633032477359559727269505708, −6.38274637281733637875714355623, −5.42704133323961923424439475452, −4.82066234745612713675703401916, −3.80569675637018367650430107868, −0.20763361549790341593987244868,
1.55035418268486757192659673714, 2.69268092309375445616880933042, 4.01857812999223005696508685715, 5.03179047291212372492823053251, 6.30923522021274422826321453350, 7.57302565341247696271702815619, 8.604273048918712490604983557811, 9.821231225313953241721189115337, 10.49554584806367700799108202517, 11.34513286025109972425122940433