| L(s) = 1 | + (1.39 − 0.258i)2-s + (−1.55 + 2.47i)3-s + (1.86 − 0.719i)4-s + (−2.53 + 2.02i)5-s + (−1.51 + 3.83i)6-s + (2.96 − 0.675i)7-s + (2.40 − 1.48i)8-s + (−2.39 − 4.96i)9-s + (−3.00 + 3.46i)10-s + (0.356 + 1.01i)11-s + (−1.11 + 5.72i)12-s + (1.93 − 4.02i)13-s + (3.94 − 1.70i)14-s + (−1.05 − 9.40i)15-s + (2.96 − 2.68i)16-s + (−1.15 + 1.15i)17-s + ⋯ |
| L(s) = 1 | + (0.983 − 0.183i)2-s + (−0.896 + 1.42i)3-s + (0.932 − 0.359i)4-s + (−1.13 + 0.904i)5-s + (−0.619 + 1.56i)6-s + (1.11 − 0.255i)7-s + (0.851 − 0.524i)8-s + (−0.797 − 1.65i)9-s + (−0.949 + 1.09i)10-s + (0.107 + 0.307i)11-s + (−0.322 + 1.65i)12-s + (0.537 − 1.11i)13-s + (1.05 − 0.455i)14-s + (−0.273 − 2.42i)15-s + (0.740 − 0.671i)16-s + (−0.279 + 0.279i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.466 - 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.466 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.13784 + 0.686142i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.13784 + 0.686142i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.39 + 0.258i)T \) |
| 29 | \( 1 + (-4.63 - 2.74i)T \) |
| good | 3 | \( 1 + (1.55 - 2.47i)T + (-1.30 - 2.70i)T^{2} \) |
| 5 | \( 1 + (2.53 - 2.02i)T + (1.11 - 4.87i)T^{2} \) |
| 7 | \( 1 + (-2.96 + 0.675i)T + (6.30 - 3.03i)T^{2} \) |
| 11 | \( 1 + (-0.356 - 1.01i)T + (-8.60 + 6.85i)T^{2} \) |
| 13 | \( 1 + (-1.93 + 4.02i)T + (-8.10 - 10.1i)T^{2} \) |
| 17 | \( 1 + (1.15 - 1.15i)T - 17iT^{2} \) |
| 19 | \( 1 + (-0.711 + 0.447i)T + (8.24 - 17.1i)T^{2} \) |
| 23 | \( 1 + (4.45 + 3.55i)T + (5.11 + 22.4i)T^{2} \) |
| 31 | \( 1 + (0.542 - 4.81i)T + (-30.2 - 6.89i)T^{2} \) |
| 37 | \( 1 + (4.22 + 1.47i)T + (28.9 + 23.0i)T^{2} \) |
| 41 | \( 1 + (3.70 + 3.70i)T + 41iT^{2} \) |
| 43 | \( 1 + (8.74 - 0.985i)T + (41.9 - 9.56i)T^{2} \) |
| 47 | \( 1 + (-6.77 + 2.37i)T + (36.7 - 29.3i)T^{2} \) |
| 53 | \( 1 + (-1.23 - 1.55i)T + (-11.7 + 51.6i)T^{2} \) |
| 59 | \( 1 + 3.13iT - 59T^{2} \) |
| 61 | \( 1 + (1.32 + 0.832i)T + (26.4 + 54.9i)T^{2} \) |
| 67 | \( 1 + (1.37 - 0.661i)T + (41.7 - 52.3i)T^{2} \) |
| 71 | \( 1 + (-3.81 - 1.83i)T + (44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (2.65 - 0.298i)T + (71.1 - 16.2i)T^{2} \) |
| 79 | \( 1 + (-15.4 - 5.40i)T + (61.7 + 49.2i)T^{2} \) |
| 83 | \( 1 + (13.0 + 2.97i)T + (74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (7.67 + 0.865i)T + (86.7 + 19.8i)T^{2} \) |
| 97 | \( 1 + (-5.71 + 3.59i)T + (42.0 - 87.3i)T^{2} \) |
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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
−14.08417820321729214854626487759, −12.27742356135436703831068845727, −11.53520632422763748334987808667, −10.66725283166844392838873111970, −10.40835804616222473838458122907, −8.185526509904946928573730270104, −6.77896089826805512900189660456, −5.35830707293881855841986810087, −4.36679266233475428588334106222, −3.43085801492675414191025814484,
1.64037827685090768467811076743, 4.26847616449630967692902375841, 5.37241171405053848158749172231, 6.58255485935251264928014713604, 7.73091397966490653604102682366, 8.398532011271412315700512118806, 11.17199947807592574723503146183, 11.85547970435662900413520367809, 12.01196648367217512753510618138, 13.32653341326090636905790253916
