L(s) = 1 | + (0.717 + 1.21i)2-s + (1.11 − 1.32i)3-s + (−0.970 + 1.74i)4-s + (1.20 + 0.323i)5-s + (2.41 + 0.407i)6-s + (0.140 − 0.242i)7-s + (−2.82 + 0.0728i)8-s + (−0.513 − 2.95i)9-s + (0.471 + 1.70i)10-s + (−3.07 + 0.823i)11-s + (1.23 + 3.23i)12-s + (2.76 + 0.740i)13-s + (0.396 − 0.00340i)14-s + (1.77 − 1.23i)15-s + (−2.11 − 3.39i)16-s + 3.72i·17-s + ⋯ |
L(s) = 1 | + (0.507 + 0.861i)2-s + (0.643 − 0.765i)3-s + (−0.485 + 0.874i)4-s + (0.539 + 0.144i)5-s + (0.986 + 0.166i)6-s + (0.0530 − 0.0918i)7-s + (−0.999 + 0.0257i)8-s + (−0.171 − 0.985i)9-s + (0.149 + 0.538i)10-s + (−0.926 + 0.248i)11-s + (0.356 + 0.934i)12-s + (0.766 + 0.205i)13-s + (0.106 − 0.000910i)14-s + (0.457 − 0.319i)15-s + (−0.529 − 0.848i)16-s + 0.902i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.773 - 0.633i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.773 - 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.52517 + 0.545027i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.52517 + 0.545027i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.717 - 1.21i)T \) |
| 3 | \( 1 + (-1.11 + 1.32i)T \) |
good | 5 | \( 1 + (-1.20 - 0.323i)T + (4.33 + 2.5i)T^{2} \) |
| 7 | \( 1 + (-0.140 + 0.242i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (3.07 - 0.823i)T + (9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (-2.76 - 0.740i)T + (11.2 + 6.5i)T^{2} \) |
| 17 | \( 1 - 3.72iT - 17T^{2} \) |
| 19 | \( 1 + (4.10 + 4.10i)T + 19iT^{2} \) |
| 23 | \( 1 + (1.57 - 0.909i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.83 - 1.02i)T + (25.1 - 14.5i)T^{2} \) |
| 31 | \( 1 + (-8.81 + 5.08i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.76 - 1.76i)T + 37iT^{2} \) |
| 41 | \( 1 + (-2.66 - 4.62i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.84 + 6.89i)T + (-37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (5.48 - 9.50i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-8.58 + 8.58i)T - 53iT^{2} \) |
| 59 | \( 1 + (-1.44 + 5.38i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-1.66 - 6.23i)T + (-52.8 + 30.5i)T^{2} \) |
| 67 | \( 1 + (1.00 - 3.75i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 10.6iT - 71T^{2} \) |
| 73 | \( 1 - 9.30iT - 73T^{2} \) |
| 79 | \( 1 + (-8.70 - 5.02i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.588 - 2.19i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + 1.87T + 89T^{2} \) |
| 97 | \( 1 + (-9.19 + 15.9i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−13.25913837693333025806847779111, −12.83416819984307581858001442810, −11.46475590433695209097444753586, −9.886496719268809925379094774928, −8.618448019376599963442614499034, −7.892189897042733309053162850928, −6.67766862379236346189904287698, −5.84668703129076994385531437093, −4.13023218686312918884492132247, −2.49396301961513652882474500168,
2.25950116544637645280007566158, 3.59231089438554220361987539890, 4.92504362854805890285563233052, 5.95851706323908153942869329750, 8.115676090996114781225337501129, 9.114839679219838044694421256596, 10.14666617792131257731041738561, 10.73542503789727903786387014969, 11.96871221059801911557271624215, 13.32465320683316334447019975124