L(s) = 1 | + (−1.20 − 2.09i)3-s + (0.5 − 0.866i)5-s + (−1.41 + 2.44i)9-s + (0.5 + 0.866i)11-s − 0.414·13-s − 2.41·15-s + (−1.20 − 2.09i)17-s + (−1 + 1.73i)19-s + (−3.12 + 5.40i)23-s + (−0.499 − 0.866i)25-s − 0.414·27-s + 29-s + (−5.12 − 8.87i)31-s + (1.20 − 2.09i)33-s + (−5.94 + 10.3i)37-s + ⋯ |
L(s) = 1 | + (−0.696 − 1.20i)3-s + (0.223 − 0.387i)5-s + (−0.471 + 0.816i)9-s + (0.150 + 0.261i)11-s − 0.114·13-s − 0.623·15-s + (−0.292 − 0.507i)17-s + (−0.229 + 0.397i)19-s + (−0.650 + 1.12i)23-s + (−0.0999 − 0.173i)25-s − 0.0797·27-s + 0.185·29-s + (−0.919 − 1.59i)31-s + (0.210 − 0.363i)33-s + (−0.978 + 1.69i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.03882525152\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.03882525152\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (1.20 + 2.09i)T + (-1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 0.414T + 13T^{2} \) |
| 17 | \( 1 + (1.20 + 2.09i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.12 - 5.40i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 + (5.12 + 8.87i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (5.94 - 10.3i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 4.58T + 41T^{2} \) |
| 43 | \( 1 + 11.6T + 43T^{2} \) |
| 47 | \( 1 + (3.79 - 6.56i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3.29 + 5.70i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.878 + 1.52i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.41 + 5.91i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.707 - 1.22i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 2.48T + 71T^{2} \) |
| 73 | \( 1 + (-5.41 - 9.37i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.67 + 2.89i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 + (4.82 - 8.36i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 14.0T + 97T^{2} \) |
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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
−9.531752987327477292014290142436, −8.392298909377652715306775368734, −7.74967386511403776232843526125, −6.97097820976062022044695147402, −6.29398014168821624788690407880, −5.57010975786192793271425688755, −4.74472263817793747892862102571, −3.55972982862249453686778384503, −2.13119956313895907924894190081, −1.36494022191740820405549841537,
0.01514302682623437574153683483, 1.95598400487570619758314104402, 3.26983867463519943360798286105, 4.08485258834485946496757842160, 4.90643311536990496119454909788, 5.64763767809327147424584274986, 6.44497068591366004737415805289, 7.21743507195144389196716702340, 8.449910923433796645479309620313, 9.041598818406488925233335792371