L(s) = 1 | + (−0.570 + 1.45i)2-s + (1.65 + 0.515i)3-s + (−0.321 − 0.298i)4-s + (2.55 − 1.22i)5-s + (−1.69 + 2.10i)6-s + (2.20 − 1.46i)7-s + (−2.19 + 1.05i)8-s + (2.46 + 1.70i)9-s + (0.330 + 4.40i)10-s + (−2.88 − 3.61i)11-s + (−0.378 − 0.659i)12-s + (−1.70 + 4.33i)13-s + (0.876 + 4.03i)14-s + (4.85 − 0.715i)15-s + (−0.350 − 4.67i)16-s + (4.28 − 3.97i)17-s + ⋯ |
L(s) = 1 | + (−0.403 + 1.02i)2-s + (0.954 + 0.297i)3-s + (−0.160 − 0.149i)4-s + (1.14 − 0.549i)5-s + (−0.691 + 0.861i)6-s + (0.832 − 0.554i)7-s + (−0.776 + 0.373i)8-s + (0.822 + 0.568i)9-s + (0.104 + 1.39i)10-s + (−0.869 − 1.08i)11-s + (−0.109 − 0.190i)12-s + (−0.471 + 1.20i)13-s + (0.234 + 1.07i)14-s + (1.25 − 0.184i)15-s + (−0.0875 − 1.16i)16-s + (1.04 − 0.965i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.221 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.221 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.52633 + 1.21872i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.52633 + 1.21872i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.65 - 0.515i)T \) |
| 7 | \( 1 + (-2.20 + 1.46i)T \) |
good | 2 | \( 1 + (0.570 - 1.45i)T + (-1.46 - 1.36i)T^{2} \) |
| 5 | \( 1 + (-2.55 + 1.22i)T + (3.11 - 3.90i)T^{2} \) |
| 11 | \( 1 + (2.88 + 3.61i)T + (-2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (1.70 - 4.33i)T + (-9.52 - 8.84i)T^{2} \) |
| 17 | \( 1 + (-4.28 + 3.97i)T + (1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (0.412 - 0.714i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.126 - 0.552i)T + (-20.7 - 9.97i)T^{2} \) |
| 29 | \( 1 + (2.40 + 0.741i)T + (23.9 + 16.3i)T^{2} \) |
| 31 | \( 1 + (3.94 - 6.83i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.13 + 0.349i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (10.1 + 6.90i)T + (14.9 + 38.1i)T^{2} \) |
| 43 | \( 1 + (0.809 - 0.551i)T + (15.7 - 40.0i)T^{2} \) |
| 47 | \( 1 + (2.96 - 7.56i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (5.50 - 1.69i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (5.21 - 3.55i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (-3.86 + 3.58i)T + (4.55 - 60.8i)T^{2} \) |
| 67 | \( 1 + (3.39 - 5.87i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.13 + 4.96i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-2.74 - 0.413i)T + (69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (-3.90 - 6.75i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.49 + 8.89i)T + (-60.8 + 56.4i)T^{2} \) |
| 89 | \( 1 + (-4.37 - 11.1i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (-2.19 + 3.79i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.11039619594470606761835904315, −10.02262464026902966757121937205, −9.225315324468298366899257308596, −8.554994908105244633601776884174, −7.73209754150137851398250808748, −6.96888744689688428848270911555, −5.58142427276032796536524197715, −4.89407540797101943139797370078, −3.19706218363182718577220988489, −1.81137885114042506951169451891,
1.75179447290671527233856786024, 2.31461273842977329927123571081, 3.32411922107453933220801443812, 5.16650686470487224413967057878, 6.22002085337674781139564027257, 7.54232031817328408990334701011, 8.316086763587529895417480408532, 9.485014268668854135805017649642, 10.09719891374499684961652007660, 10.53636488671439408868915994293