L(s) = 1 | − 1.10·2-s + (1.22 + 1.22i)3-s − 0.783·4-s + (0.0527 + 0.0913i)5-s + (−1.34 − 1.35i)6-s + 3.07·8-s + (−0.0231 + 2.99i)9-s + (−0.0581 − 0.100i)10-s + (−1.66 + 2.89i)11-s + (−0.956 − 0.963i)12-s + (1.23 − 2.14i)13-s + (−0.0479 + 0.176i)15-s − 1.81·16-s + (0.806 + 1.39i)17-s + (0.0255 − 3.30i)18-s + (−3.84 + 6.65i)19-s + ⋯ |
L(s) = 1 | − 0.779·2-s + (0.704 + 0.709i)3-s − 0.391·4-s + (0.0235 + 0.0408i)5-s + (−0.549 − 0.553i)6-s + 1.08·8-s + (−0.00772 + 0.999i)9-s + (−0.0183 − 0.0318i)10-s + (−0.503 + 0.871i)11-s + (−0.276 − 0.278i)12-s + (0.343 − 0.595i)13-s + (−0.0123 + 0.0455i)15-s − 0.454·16-s + (0.195 + 0.338i)17-s + (0.00602 − 0.779i)18-s + (−0.881 + 1.52i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.358 - 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.358 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.495203 + 0.720269i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.495203 + 0.720269i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.22 - 1.22i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 1.10T + 2T^{2} \) |
| 5 | \( 1 + (-0.0527 - 0.0913i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1.66 - 2.89i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.23 + 2.14i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.806 - 1.39i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.84 - 6.65i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.948 - 1.64i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.64 - 8.04i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 9.26T + 31T^{2} \) |
| 37 | \( 1 + (-0.991 + 1.71i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.74 - 6.48i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.77 + 6.53i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 3.19T + 47T^{2} \) |
| 53 | \( 1 + (-4.98 - 8.64i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 4.45T + 59T^{2} \) |
| 61 | \( 1 - 5.67T + 61T^{2} \) |
| 67 | \( 1 - 9.97T + 67T^{2} \) |
| 71 | \( 1 - 3.29T + 71T^{2} \) |
| 73 | \( 1 + (2.36 + 4.09i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 7.69T + 79T^{2} \) |
| 83 | \( 1 + (-0.584 - 1.01i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-3.01 + 5.22i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.90 - 3.29i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.66451185923640056983786996215, −10.45111041193054090819150750825, −9.611586835880811751737834442458, −8.642890522022771763009300899157, −8.131514936896657907122529920623, −7.18715162088519975289141705352, −5.50628518420873452113043634121, −4.50892013998973544944895810626, −3.46938002740153205563953806598, −1.84439540141156309415139255101,
0.68307751526383457762132416825, 2.28621815287222489669552076324, 3.70530384170850885522536834974, 5.03792783947039458993475962350, 6.49279375151158784901223527698, 7.37674579888917745812958116195, 8.365277138778092199145104577604, 8.860491700012788579034408404004, 9.610582931898618015161928296291, 10.77315222089390762820768508365