L(s) = 1 | + (1.65 + 0.578i)2-s + (−0.186 + 1.65i)3-s + (−0.727 − 0.579i)4-s + (−0.825 − 1.71i)5-s + (−1.26 + 2.62i)6-s + (−1.24 − 1.56i)7-s + (−4.59 − 7.31i)8-s + (6.07 + 1.38i)9-s + (−0.373 − 3.31i)10-s + (−6.63 + 10.5i)11-s + (1.09 − 1.09i)12-s + (−3.80 + 0.868i)13-s + (−1.15 − 3.30i)14-s + (2.98 − 1.04i)15-s + (−2.54 − 11.1i)16-s + (7.59 + 7.59i)17-s + ⋯ |
L(s) = 1 | + (0.826 + 0.289i)2-s + (−0.0621 + 0.551i)3-s + (−0.181 − 0.144i)4-s + (−0.165 − 0.342i)5-s + (−0.210 + 0.437i)6-s + (−0.178 − 0.223i)7-s + (−0.574 − 0.914i)8-s + (0.674 + 0.154i)9-s + (−0.0373 − 0.331i)10-s + (−0.602 + 0.959i)11-s + (0.0912 − 0.0912i)12-s + (−0.292 + 0.0668i)13-s + (−0.0827 − 0.236i)14-s + (0.199 − 0.0697i)15-s + (−0.158 − 0.695i)16-s + (0.446 + 0.446i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 - 0.407i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.913 - 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.20315 + 0.256245i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.20315 + 0.256245i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 29 | \( 1 + (-8.15 + 27.8i)T \) |
good | 2 | \( 1 + (-1.65 - 0.578i)T + (3.12 + 2.49i)T^{2} \) |
| 3 | \( 1 + (0.186 - 1.65i)T + (-8.77 - 2.00i)T^{2} \) |
| 5 | \( 1 + (0.825 + 1.71i)T + (-15.5 + 19.5i)T^{2} \) |
| 7 | \( 1 + (1.24 + 1.56i)T + (-10.9 + 47.7i)T^{2} \) |
| 11 | \( 1 + (6.63 - 10.5i)T + (-52.4 - 109. i)T^{2} \) |
| 13 | \( 1 + (3.80 - 0.868i)T + (152. - 73.3i)T^{2} \) |
| 17 | \( 1 + (-7.59 - 7.59i)T + 289iT^{2} \) |
| 19 | \( 1 + (0.137 - 0.0154i)T + (351. - 80.3i)T^{2} \) |
| 23 | \( 1 + (-26.7 - 12.8i)T + (329. + 413. i)T^{2} \) |
| 31 | \( 1 + (54.1 + 18.9i)T + (751. + 599. i)T^{2} \) |
| 37 | \( 1 + (29.9 + 47.6i)T + (-593. + 1.23e3i)T^{2} \) |
| 41 | \( 1 + (25.9 - 25.9i)T - 1.68e3iT^{2} \) |
| 43 | \( 1 + (-5.16 - 14.7i)T + (-1.44e3 + 1.15e3i)T^{2} \) |
| 47 | \( 1 + (-55.9 - 35.1i)T + (958. + 1.99e3i)T^{2} \) |
| 53 | \( 1 + (-29.3 + 14.1i)T + (1.75e3 - 2.19e3i)T^{2} \) |
| 59 | \( 1 - 0.396T + 3.48e3T^{2} \) |
| 61 | \( 1 + (6.99 - 62.1i)T + (-3.62e3 - 828. i)T^{2} \) |
| 67 | \( 1 + (21.7 + 4.97i)T + (4.04e3 + 1.94e3i)T^{2} \) |
| 71 | \( 1 + (64.8 - 14.7i)T + (4.54e3 - 2.18e3i)T^{2} \) |
| 73 | \( 1 + (-76.3 + 26.7i)T + (4.16e3 - 3.32e3i)T^{2} \) |
| 79 | \( 1 + (-67.7 + 42.5i)T + (2.70e3 - 5.62e3i)T^{2} \) |
| 83 | \( 1 + (68.4 - 85.8i)T + (-1.53e3 - 6.71e3i)T^{2} \) |
| 89 | \( 1 + (77.7 + 27.2i)T + (6.19e3 + 4.93e3i)T^{2} \) |
| 97 | \( 1 + (-19.5 - 173. i)T + (-9.17e3 + 2.09e3i)T^{2} \) |
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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
−16.62733296614835653656893795338, −15.51992463722839861181769819303, −14.75369997801927775411263153892, −13.25003498915606651007064915554, −12.45603964869917797688440304243, −10.43968157939932586521769603215, −9.370494919291570751956801127243, −7.24356589831141414919023073603, −5.28535564427513188446412866702, −4.10954497117774731565483413246,
3.21028728649660128859876654384, 5.26814537954540571231719335186, 7.14139242802598570942609641989, 8.806964114095973891357057442634, 10.78736868471982241225272529820, 12.20821867456442800486865002371, 13.02863400302361874599003254202, 14.09393946313299493877267908459, 15.35744191249550161358643275556, 16.83753293767611771842095039381