L(s) = 1 | + (782. − 782. i)2-s + (−1.28e4 − 1.28e4i)3-s − 1.77e5i·4-s + (7.95e6 + 5.66e6i)5-s − 2.00e7·6-s + (3.15e8 − 3.15e8i)7-s + (6.82e8 + 6.82e8i)8-s − 3.15e9i·9-s + (1.06e10 − 1.79e9i)10-s − 2.67e10·11-s + (−2.26e9 + 2.26e9i)12-s + (9.28e9 + 9.28e9i)13-s − 4.93e11i·14-s + (−2.93e10 − 1.74e11i)15-s + 1.25e12·16-s + (2.62e12 − 2.62e12i)17-s + ⋯ |
L(s) = 1 | + (0.764 − 0.764i)2-s + (−0.216 − 0.216i)3-s − 0.168i·4-s + (0.814 + 0.580i)5-s − 0.331·6-s + (1.11 − 1.11i)7-s + (0.635 + 0.635i)8-s − 0.906i·9-s + (1.06 − 0.179i)10-s − 1.03·11-s + (−0.0366 + 0.0366i)12-s + (0.0673 + 0.0673i)13-s − 1.70i·14-s + (−0.0508 − 0.302i)15-s + 1.14·16-s + (1.30 − 1.30i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.387 + 0.921i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (0.387 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{21}{2})\) |
\(\approx\) |
\(2.60878 - 1.73229i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.60878 - 1.73229i\) |
\(L(11)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-7.95e6 - 5.66e6i)T \) |
good | 2 | \( 1 + (-782. + 782. i)T - 1.04e6iT^{2} \) |
| 3 | \( 1 + (1.28e4 + 1.28e4i)T + 3.48e9iT^{2} \) |
| 7 | \( 1 + (-3.15e8 + 3.15e8i)T - 7.97e16iT^{2} \) |
| 11 | \( 1 + 2.67e10T + 6.72e20T^{2} \) |
| 13 | \( 1 + (-9.28e9 - 9.28e9i)T + 1.90e22iT^{2} \) |
| 17 | \( 1 + (-2.62e12 + 2.62e12i)T - 4.06e24iT^{2} \) |
| 19 | \( 1 - 1.22e12iT - 3.75e25T^{2} \) |
| 23 | \( 1 + (-1.74e13 - 1.74e13i)T + 1.71e27iT^{2} \) |
| 29 | \( 1 - 2.30e13iT - 1.76e29T^{2} \) |
| 31 | \( 1 + 9.47e13T + 6.71e29T^{2} \) |
| 37 | \( 1 + (3.89e15 - 3.89e15i)T - 2.31e31iT^{2} \) |
| 41 | \( 1 + 2.23e16T + 1.80e32T^{2} \) |
| 43 | \( 1 + (-1.30e16 - 1.30e16i)T + 4.67e32iT^{2} \) |
| 47 | \( 1 + (1.21e16 - 1.21e16i)T - 2.76e33iT^{2} \) |
| 53 | \( 1 + (9.59e16 + 9.59e16i)T + 3.05e34iT^{2} \) |
| 59 | \( 1 - 9.18e17iT - 2.61e35T^{2} \) |
| 61 | \( 1 + 6.23e16T + 5.08e35T^{2} \) |
| 67 | \( 1 + (4.75e17 - 4.75e17i)T - 3.32e36iT^{2} \) |
| 71 | \( 1 - 2.91e18T + 1.05e37T^{2} \) |
| 73 | \( 1 + (3.68e18 + 3.68e18i)T + 1.84e37iT^{2} \) |
| 79 | \( 1 + 2.24e18iT - 8.96e37T^{2} \) |
| 83 | \( 1 + (7.91e18 + 7.91e18i)T + 2.40e38iT^{2} \) |
| 89 | \( 1 - 1.81e18iT - 9.72e38T^{2} \) |
| 97 | \( 1 + (-2.60e19 + 2.60e19i)T - 5.43e39iT^{2} \) |
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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
−18.29038363689814070400441077181, −17.11636694956027563568950732537, −14.44090841452566555812386019717, −13.41170505774397745411536425876, −11.67161426258776424702774509369, −10.33089645499576554045002567044, −7.45643317735524370398309379343, −5.12638268766302884432384910395, −3.18076165367505940583305706662, −1.35579833420532458318197378637,
1.81168490950033558138649206795, 5.01567745325729183638105029796, 5.62475522940624530613305983578, 8.174468327556595861677551135719, 10.44317817484210961932388973771, 12.74826156560639813999917852378, 14.22810855171721596211287393439, 15.57020279465654029394110079968, 17.01553519489451870133355480109, 18.74362162285391046810150328088