L(s) = 1 | + (1.96e4 + 5.56e4i)3-s − 7.88e6i·5-s + 3.14e8·7-s + (−2.71e9 + 2.18e9i)9-s − 3.97e10i·11-s − 2.60e11·13-s + (4.39e11 − 1.54e11i)15-s + 4.22e10i·17-s + 1.21e12·19-s + (6.17e12 + 1.75e13i)21-s − 4.94e13i·23-s + 3.31e13·25-s + (−1.75e14 − 1.08e14i)27-s − 4.36e14i·29-s + 3.04e14·31-s + ⋯ |
L(s) = 1 | + (0.332 + 0.943i)3-s − 0.807i·5-s + 1.11·7-s + (−0.779 + 0.626i)9-s − 1.53i·11-s − 1.89·13-s + (0.761 − 0.268i)15-s + 0.0209i·17-s + 0.198·19-s + (0.369 + 1.05i)21-s − 1.19i·23-s + 0.348·25-s + (−0.850 − 0.526i)27-s − 1.03i·29-s + 0.371·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.332 + 0.943i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (0.332 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{21}{2})\) |
\(\approx\) |
\(1.722779750\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.722779750\) |
\(L(11)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.96e4 - 5.56e4i)T \) |
good | 5 | \( 1 + 7.88e6iT - 9.53e13T^{2} \) |
| 7 | \( 1 - 3.14e8T + 7.97e16T^{2} \) |
| 11 | \( 1 + 3.97e10iT - 6.72e20T^{2} \) |
| 13 | \( 1 + 2.60e11T + 1.90e22T^{2} \) |
| 17 | \( 1 - 4.22e10iT - 4.06e24T^{2} \) |
| 19 | \( 1 - 1.21e12T + 3.75e25T^{2} \) |
| 23 | \( 1 + 4.94e13iT - 1.71e27T^{2} \) |
| 29 | \( 1 + 4.36e14iT - 1.76e29T^{2} \) |
| 31 | \( 1 - 3.04e14T + 6.71e29T^{2} \) |
| 37 | \( 1 - 1.82e15T + 2.31e31T^{2} \) |
| 41 | \( 1 + 2.29e16iT - 1.80e32T^{2} \) |
| 43 | \( 1 - 1.37e16T + 4.67e32T^{2} \) |
| 47 | \( 1 + 4.50e16iT - 2.76e33T^{2} \) |
| 53 | \( 1 - 1.96e17iT - 3.05e34T^{2} \) |
| 59 | \( 1 - 4.53e17iT - 2.61e35T^{2} \) |
| 61 | \( 1 - 1.15e17T + 5.08e35T^{2} \) |
| 67 | \( 1 - 7.02e17T + 3.32e36T^{2} \) |
| 71 | \( 1 + 4.29e18iT - 1.05e37T^{2} \) |
| 73 | \( 1 + 7.58e18T + 1.84e37T^{2} \) |
| 79 | \( 1 - 1.37e19T + 8.96e37T^{2} \) |
| 83 | \( 1 - 9.70e18iT - 2.40e38T^{2} \) |
| 89 | \( 1 - 1.74e18iT - 9.72e38T^{2} \) |
| 97 | \( 1 + 1.87e19T + 5.43e39T^{2} \) |
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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
−14.91890692735435999708547704093, −13.89606034809589167956103912861, −11.98196663640804679868148603538, −10.60361280134484344984579225214, −9.041270322927227056675923991156, −7.999452846221085664639183192355, −5.37892058146146291869100120963, −4.39100282481864034419067828339, −2.52466835753088437537607539666, −0.50391776932807223405212972725,
1.58799641841920536786058420123, 2.68453808917971367830837421141, 4.91019485945021819773781449652, 7.00103070522047887866240174030, 7.74428668061471525170786985312, 9.713191817878722121195054797323, 11.49949194116428416731069646615, 12.64607897165756176686635733431, 14.44218380243875932675537164431, 14.87290547709257357733076857745