L(s) = 1 | + (0.327 + 1.37i)2-s + (1.42 − 2.47i)3-s + (−1.78 + 0.902i)4-s + (−0.139 + 0.242i)5-s + (3.86 + 1.15i)6-s + 1.55i·7-s + (−1.82 − 2.15i)8-s + (−2.57 − 4.45i)9-s + (−0.379 − 0.113i)10-s + 2.44i·11-s + (−0.317 + 5.69i)12-s + (−5.47 + 3.16i)13-s + (−2.13 + 0.509i)14-s + (0.399 + 0.691i)15-s + (2.37 − 3.22i)16-s + (2.18 − 3.78i)17-s + ⋯ |
L(s) = 1 | + (0.231 + 0.972i)2-s + (0.823 − 1.42i)3-s + (−0.892 + 0.451i)4-s + (−0.0625 + 0.108i)5-s + (1.57 + 0.470i)6-s + 0.586i·7-s + (−0.645 − 0.763i)8-s + (−0.857 − 1.48i)9-s + (−0.119 − 0.0357i)10-s + 0.735i·11-s + (−0.0916 + 1.64i)12-s + (−1.51 + 0.876i)13-s + (−0.570 + 0.136i)14-s + (0.103 + 0.178i)15-s + (0.593 − 0.805i)16-s + (0.529 − 0.917i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.11969 + 0.188367i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.11969 + 0.188367i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.327 - 1.37i)T \) |
| 19 | \( 1 + (3.17 + 2.99i)T \) |
good | 3 | \( 1 + (-1.42 + 2.47i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (0.139 - 0.242i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 - 1.55iT - 7T^{2} \) |
| 11 | \( 1 - 2.44iT - 11T^{2} \) |
| 13 | \( 1 + (5.47 - 3.16i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.18 + 3.78i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-5.71 + 3.30i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.695 + 0.401i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 1.42T + 31T^{2} \) |
| 37 | \( 1 - 4.74iT - 37T^{2} \) |
| 41 | \( 1 + (5.84 + 3.37i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.146 - 0.0845i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.19 + 0.691i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.25 + 2.45i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.05 + 1.81i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.58 + 2.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.00 + 1.73i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.74 - 3.02i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (5.43 - 9.41i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.00 + 5.19i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 9.99iT - 83T^{2} \) |
| 89 | \( 1 + (-7.76 + 4.48i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.77 + 1.02i)T + (48.5 + 84.0i)T^{2} \) |
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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
−14.65820264813378889729668754019, −13.62288683223872209893459646074, −12.65097668727785047655869118445, −11.97643968720825827874849009999, −9.477094371316353317525878463644, −8.598732637951076112049110880117, −7.22449682409080177905038991907, −6.88156930986466182722791626002, −4.96763409663635650445370529313, −2.64016570664415265477930387265,
2.97600718015399579114084827220, 4.10921366260136098055817218984, 5.32790007631649568049406409426, 8.065853514864967524085005723884, 9.141694713483646035651364796910, 10.27871600839276948970988226463, 10.65722846647068491855529803316, 12.28116141299503303133318877826, 13.45610383497322130485723152045, 14.63649094529354955445997723995