L(s) = 1 | + (−1.08 + 1.88i)2-s + (1.68 − 0.388i)3-s + (−1.36 − 2.36i)4-s + (−0.634 − 1.09i)5-s + (−1.10 + 3.60i)6-s + 1.60·8-s + (2.69 − 1.31i)9-s + 2.76·10-s + (2.73 − 4.74i)11-s + (−3.23 − 3.46i)12-s + (2.37 + 4.10i)13-s + (−1.49 − 1.60i)15-s + (0.992 − 1.71i)16-s − 4.81·17-s + (−0.463 + 6.51i)18-s + 5.38·19-s + ⋯ |
L(s) = 1 | + (−0.769 + 1.33i)2-s + (0.974 − 0.224i)3-s + (−0.684 − 1.18i)4-s + (−0.283 − 0.491i)5-s + (−0.450 + 1.47i)6-s + 0.566·8-s + (0.899 − 0.437i)9-s + 0.872·10-s + (0.825 − 1.43i)11-s + (−0.932 − 1.00i)12-s + (0.658 + 1.13i)13-s + (−0.386 − 0.415i)15-s + (0.248 − 0.429i)16-s − 1.16·17-s + (−0.109 + 1.53i)18-s + 1.23·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.695 - 0.718i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.695 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.18260 + 0.501097i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.18260 + 0.501097i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.68 + 0.388i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (1.08 - 1.88i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (0.634 + 1.09i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.73 + 4.74i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.37 - 4.10i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 4.81T + 17T^{2} \) |
| 19 | \( 1 - 5.38T + 19T^{2} \) |
| 23 | \( 1 + (-2.58 - 4.48i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.01 + 3.49i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.732 - 1.26i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 1.91T + 37T^{2} \) |
| 41 | \( 1 + (1.94 + 3.37i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.66 - 2.87i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.57 - 2.73i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 7.14T + 53T^{2} \) |
| 59 | \( 1 + (0.154 + 0.267i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.17 - 8.95i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.23 + 3.87i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 1.96T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 + (-4.50 + 7.80i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (5.08 - 8.79i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 5.19T + 89T^{2} \) |
| 97 | \( 1 + (2.48 - 4.30i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.22592197976994729952055746142, −9.681217622856890562596678550153, −8.896798363548701652372105194188, −8.670220915353595525284759257202, −7.68994453273283311977160040717, −6.76100144495595726396300693396, −6.04617099656342813794849744488, −4.51219232036641048179522868664, −3.26053224816898564224477424306, −1.16112097932892923654049765950,
1.47687046560232978014272727038, 2.74465740423828460634961363500, 3.53960808643213291753063688137, 4.69102645269590234892293248626, 6.70248824568206736046591804823, 7.66604687819454814206512696972, 8.672024262707258514932086862558, 9.331542748772976153894423214946, 10.12926038269411312484803471216, 10.79622381786692982285015295326