L(s) = 1 | + (1.27 + 2.20i)2-s + (−2.25 + 3.90i)4-s + (1.39 − 2.41i)5-s + (2.06 − 1.64i)7-s − 6.39·8-s + 7.10·10-s + 2.76·11-s + (2.99 − 2.01i)13-s + (6.28 + 2.46i)14-s + (−3.64 − 6.31i)16-s + (−2.94 + 5.09i)17-s + 3.40·19-s + (6.27 + 10.8i)20-s + (3.52 + 6.11i)22-s + (−3.67 − 6.36i)23-s + ⋯ |
L(s) = 1 | + (0.901 + 1.56i)2-s + (−1.12 + 1.95i)4-s + (0.623 − 1.07i)5-s + (0.782 − 0.623i)7-s − 2.25·8-s + 2.24·10-s + 0.834·11-s + (0.830 − 0.557i)13-s + (1.67 + 0.659i)14-s + (−0.910 − 1.57i)16-s + (−0.713 + 1.23i)17-s + 0.780·19-s + (1.40 + 2.43i)20-s + (0.752 + 1.30i)22-s + (−0.765 − 1.32i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0121 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0121 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.02564 + 2.05031i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.02564 + 2.05031i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-2.06 + 1.64i)T \) |
| 13 | \( 1 + (-2.99 + 2.01i)T \) |
good | 2 | \( 1 + (-1.27 - 2.20i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (-1.39 + 2.41i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 - 2.76T + 11T^{2} \) |
| 17 | \( 1 + (2.94 - 5.09i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 - 3.40T + 19T^{2} \) |
| 23 | \( 1 + (3.67 + 6.36i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.56 - 2.70i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.93 - 3.34i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.92 + 5.06i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3.24 - 5.61i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.99 - 5.18i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (3.95 - 6.85i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (6.34 + 10.9i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.36 + 2.36i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 - 9.55T + 61T^{2} \) |
| 67 | \( 1 + 8.68T + 67T^{2} \) |
| 71 | \( 1 + (-1.57 - 2.72i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (4.80 + 8.31i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.88 - 3.25i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 4.82T + 83T^{2} \) |
| 89 | \( 1 + (-0.877 - 1.52i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-8.48 - 14.6i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.37450170810255442017961242640, −9.113953041769148646129613557152, −8.430667969988484084533498977157, −7.930775384235523259520827194244, −6.70454352659246078047243111826, −6.08950679360616622983364885881, −5.16439670230326791880719702904, −4.46724011385836866493677232079, −3.64962739174125046992789558796, −1.43600376873028875432278563480,
1.53557730294854673516762020124, 2.36530249722327871942503934275, 3.35938870403834148500644678278, 4.32613781409431516639089740804, 5.41350887762355888160879359107, 6.14414655559180364960986645657, 7.24627659083395595165195040705, 8.805737418222517198028859863274, 9.530180369735330470573557308164, 10.22436770966274696954078120283