| L(s) = 1 | + (−0.174 + 1.98i)2-s + (−1.19 + 1.25i)3-s + (−1.95 − 0.344i)4-s + (−1.97 − 1.04i)5-s + (−2.28 − 2.59i)6-s + (−0.808 + 1.15i)7-s + (−0.00692 + 0.0258i)8-s + (−0.145 − 2.99i)9-s + (2.42 − 3.74i)10-s + (−1.21 + 3.35i)11-s + (2.76 − 2.04i)12-s + (3.03 − 0.265i)13-s + (−2.15 − 1.80i)14-s + (3.67 − 1.22i)15-s + (−3.78 − 1.37i)16-s + (1.05 + 3.91i)17-s + ⋯ |
| L(s) = 1 | + (−0.123 + 1.40i)2-s + (−0.689 + 0.724i)3-s + (−0.978 − 0.172i)4-s + (−0.883 − 0.468i)5-s + (−0.933 − 1.05i)6-s + (−0.305 + 0.436i)7-s + (−0.00244 + 0.00914i)8-s + (−0.0484 − 0.998i)9-s + (0.767 − 1.18i)10-s + (−0.367 + 1.01i)11-s + (0.799 − 0.589i)12-s + (0.842 − 0.0737i)13-s + (−0.576 − 0.483i)14-s + (0.948 − 0.316i)15-s + (−0.945 − 0.344i)16-s + (0.254 + 0.950i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 + 0.443i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.896 + 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.129078 - 0.552373i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.129078 - 0.552373i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.19 - 1.25i)T \) |
| 5 | \( 1 + (1.97 + 1.04i)T \) |
| good | 2 | \( 1 + (0.174 - 1.98i)T + (-1.96 - 0.347i)T^{2} \) |
| 7 | \( 1 + (0.808 - 1.15i)T + (-2.39 - 6.57i)T^{2} \) |
| 11 | \( 1 + (1.21 - 3.35i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-3.03 + 0.265i)T + (12.8 - 2.25i)T^{2} \) |
| 17 | \( 1 + (-1.05 - 3.91i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-1.42 + 0.821i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (5.24 - 3.67i)T + (7.86 - 21.6i)T^{2} \) |
| 29 | \( 1 + (-2.12 + 1.78i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (1.13 - 6.43i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (2.80 - 0.751i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-5.92 + 7.06i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-5.17 + 2.41i)T + (27.6 - 32.9i)T^{2} \) |
| 47 | \( 1 + (-6.52 - 4.56i)T + (16.0 + 44.1i)T^{2} \) |
| 53 | \( 1 + (8.37 + 8.37i)T + 53iT^{2} \) |
| 59 | \( 1 + (-13.4 + 4.89i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.11 - 6.32i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (1.01 + 11.6i)T + (-65.9 + 11.6i)T^{2} \) |
| 71 | \( 1 + (-0.966 - 0.557i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (8.19 + 2.19i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-3.51 - 4.18i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (6.05 + 0.529i)T + (81.7 + 14.4i)T^{2} \) |
| 89 | \( 1 + (-0.0742 - 0.128i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.45 - 11.6i)T + (-62.3 + 74.3i)T^{2} \) |
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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
−14.25807075293507079573287297270, −12.68143439120470659874448263351, −11.88125520999276698289243765954, −10.69156950764216834111072204478, −9.377115210582186566277661702008, −8.416539060839400084617282423034, −7.33395417782627608835301595751, −6.09656807801552120317413405827, −5.16695189474910174290428884414, −3.93538688846043020701175381877,
0.67074234459042994573251527939, 2.79567463397063405806686029731, 4.10833972477500518096731330024, 6.04590303106751095421465864502, 7.27353927955406393230825970416, 8.448185903565164477225525354024, 10.08145348567146955120126530987, 10.97442314053416369730410418991, 11.49238100075771869758672830613, 12.33248425015662138843594565314
