L(s) = 1 | + (−1.59 − 0.667i)3-s − 3.69·5-s + (2.60 − 0.436i)7-s + (2.10 + 2.13i)9-s + 0.892·11-s + (0.598 + 1.03i)13-s + (5.90 + 2.46i)15-s + (−0.124 − 0.216i)17-s + (−1.40 + 2.43i)19-s + (−4.46 − 1.04i)21-s − 2.47·23-s + 8.63·25-s + (−1.94 − 4.81i)27-s + (2.07 − 3.58i)29-s + (1.79 − 3.10i)31-s + ⋯ |
L(s) = 1 | + (−0.922 − 0.385i)3-s − 1.65·5-s + (0.986 − 0.165i)7-s + (0.703 + 0.711i)9-s + 0.269·11-s + (0.165 + 0.287i)13-s + (1.52 + 0.636i)15-s + (−0.0303 − 0.0525i)17-s + (−0.322 + 0.557i)19-s + (−0.973 − 0.227i)21-s − 0.516·23-s + 1.72·25-s + (−0.374 − 0.927i)27-s + (0.384 − 0.666i)29-s + (0.321 − 0.557i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.211 + 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.211 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6490845956\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6490845956\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.59 + 0.667i)T \) |
| 7 | \( 1 + (-2.60 + 0.436i)T \) |
good | 5 | \( 1 + 3.69T + 5T^{2} \) |
| 11 | \( 1 - 0.892T + 11T^{2} \) |
| 13 | \( 1 + (-0.598 - 1.03i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (0.124 + 0.216i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.40 - 2.43i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 2.47T + 23T^{2} \) |
| 29 | \( 1 + (-2.07 + 3.58i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.79 + 3.10i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.36 - 4.09i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.39 + 4.14i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.98 + 8.64i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (5.08 + 8.81i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.94 + 8.56i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.906 + 1.56i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.40 + 9.35i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.514 + 0.891i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 4.94T + 71T^{2} \) |
| 73 | \( 1 + (0.915 + 1.58i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.899 + 1.55i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (6.16 - 10.6i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (1.20 - 2.08i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.52 + 9.56i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.976481112730931313051220102574, −8.484727720060891319975331491435, −8.030850893172273717564922058024, −7.25476849041129800520514404339, −6.47197486641507875834665812274, −5.27338592480307210474199359548, −4.41374474511301566571956440515, −3.74231299982359687946564956186, −1.86355692597800837364736237315, −0.40808761554387866019422436769,
1.14748624208606932440181282019, 3.16474203234054382173576562135, 4.34564514526053702354137278137, 4.66520780119659498655002894451, 5.84788517656725778186184412164, 6.89686253986887894002678260861, 7.71230880269576290557880163016, 8.424896491969871686078507731122, 9.325300198662291174440914092186, 10.63638760600558656708036326976