L(s) = 1 | + (0.939 − 0.342i)2-s + (0.173 − 0.984i)3-s + (0.766 − 0.642i)4-s + (−0.173 − 0.984i)6-s + (0.5 + 0.866i)7-s + (0.500 − 0.866i)8-s + (1.87 + 0.684i)9-s + (3 − 5.19i)11-s + (−0.5 − 0.866i)12-s + (0.868 + 4.92i)13-s + (0.766 + 0.642i)14-s + (0.173 − 0.984i)16-s + (−2.81 + 1.02i)17-s + 2·18-s + (0.939 − 0.342i)21-s + (1.04 − 5.90i)22-s + ⋯ |
L(s) = 1 | + (0.664 − 0.241i)2-s + (0.100 − 0.568i)3-s + (0.383 − 0.321i)4-s + (−0.0708 − 0.402i)6-s + (0.188 + 0.327i)7-s + (0.176 − 0.306i)8-s + (0.626 + 0.228i)9-s + (0.904 − 1.56i)11-s + (−0.144 − 0.249i)12-s + (0.240 + 1.36i)13-s + (0.204 + 0.171i)14-s + (0.0434 − 0.246i)16-s + (−0.683 + 0.248i)17-s + 0.471·18-s + (0.205 − 0.0746i)21-s + (0.222 − 1.25i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.527 + 0.849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.527 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.23951 - 1.24621i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.23951 - 1.24621i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.939 + 0.342i)T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (-0.173 + 0.984i)T + (-2.81 - 1.02i)T^{2} \) |
| 5 | \( 1 + (0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (-0.5 - 0.866i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-3 + 5.19i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.868 - 4.92i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (2.81 - 1.02i)T + (13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (-2.29 + 1.92i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (8.45 + 3.07i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-6.12 - 5.14i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (2.29 - 1.92i)T + (9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (8.45 - 3.07i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (7.66 - 6.42i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (4.69 + 1.71i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (4.59 + 3.85i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (1.21 - 6.89i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (1.73 - 9.84i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-3 - 5.19i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (2.08 + 11.8i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-9.39 + 3.42i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.54833784764028093750055139258, −9.259260296216109981607584093033, −8.644685570103900191559711166479, −7.51467459439590255493571707344, −6.47179989868807803321139707732, −6.04686995954062140125928422911, −4.58873909472711908625287774867, −3.83427279337972830970364107300, −2.43278765033373228012444590842, −1.31200864098846274265031601308,
1.67451830837371868014513434309, 3.30010251058060033451872534468, 4.19821164256336056053636932056, 4.89989448932821614647582959161, 5.99474502098089234579135791982, 7.22286667638008259855198378332, 7.52084976166429710729298934557, 9.054577461487803479211601364300, 9.637461529366752335236881423590, 10.60809234454249016194214761944