L(s) = 1 | + (−1.31 + 2.28i)2-s + (−1.71 − 0.238i)3-s + (−2.46 − 4.27i)4-s + (2.80 − 3.59i)6-s + (0.898 − 1.55i)7-s + 7.73·8-s + (2.88 + 0.817i)9-s + (−0.904 + 1.56i)11-s + (3.21 + 7.92i)12-s + (−0.985 − 1.70i)13-s + (2.36 + 4.09i)14-s + (−5.24 + 9.08i)16-s + 4.80·17-s + (−5.66 + 5.50i)18-s + 2.96·19-s + ⋯ |
L(s) = 1 | + (−0.931 + 1.61i)2-s + (−0.990 − 0.137i)3-s + (−1.23 − 2.13i)4-s + (1.14 − 1.46i)6-s + (0.339 − 0.588i)7-s + 2.73·8-s + (0.962 + 0.272i)9-s + (−0.272 + 0.472i)11-s + (0.928 + 2.28i)12-s + (−0.273 − 0.473i)13-s + (0.632 + 1.09i)14-s + (−1.31 + 2.27i)16-s + 1.16·17-s + (−1.33 + 1.29i)18-s + 0.680·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.435 - 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.435 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.457688 + 0.287007i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.457688 + 0.287007i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.71 + 0.238i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (1.31 - 2.28i)T + (-1 - 1.73i)T^{2} \) |
| 7 | \( 1 + (-0.898 + 1.55i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.904 - 1.56i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.985 + 1.70i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 4.80T + 17T^{2} \) |
| 19 | \( 1 - 2.96T + 19T^{2} \) |
| 23 | \( 1 + (-0.866 - 1.50i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.68 + 6.38i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.31 - 2.27i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 11.6T + 37T^{2} \) |
| 41 | \( 1 + (-1.23 - 2.13i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.63 + 6.30i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3.14 - 5.44i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 1.72T + 53T^{2} \) |
| 59 | \( 1 + (5.51 + 9.54i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.33 + 10.9i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.55 + 7.88i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 1.27T + 71T^{2} \) |
| 73 | \( 1 + 3.58T + 73T^{2} \) |
| 79 | \( 1 + (1.05 - 1.82i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (0.549 - 0.951i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 + (-1.91 + 3.31i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−12.45073398662297473723352563915, −11.14256557722349340815686125877, −10.11098901435450961672655877233, −9.594100762666926478469460882361, −7.87523690292576485218778116302, −7.58126003002955178360959634788, −6.43038087437019438762412224884, −5.49698956370005502817057389730, −4.59402325899071908097051356758, −0.936095320665983510351645712985,
1.14495017717029982320766052489, 2.84906343667717751208456700928, 4.33637659323978499129268488313, 5.61906025384137945307670046960, 7.36830814677326221184383395653, 8.493300735488972511425002777456, 9.538850398801006969488140649899, 10.26279034528876549064452209897, 11.18180379242727770582966705151, 11.81837281577750647657990071300