L(s) = 1 | + (−1.69 + 1.06i)2-s + (−0.988 + 0.149i)3-s + (1.29 − 2.69i)4-s + (1.51 − 1.30i)6-s + (0.446 + 3.96i)8-s + (0.955 − 0.294i)9-s + (−0.882 + 2.86i)12-s + (−1.14 + 0.914i)13-s + (−3.10 − 3.88i)16-s + (−1.30 + 1.51i)18-s + (0.974 − 0.222i)23-s + (−1.03 − 3.85i)24-s + (−0.900 − 0.433i)25-s + (0.967 − 2.76i)26-s + (−0.900 + 0.433i)27-s + ⋯ |
L(s) = 1 | + (−1.69 + 1.06i)2-s + (−0.988 + 0.149i)3-s + (1.29 − 2.69i)4-s + (1.51 − 1.30i)6-s + (0.446 + 3.96i)8-s + (0.955 − 0.294i)9-s + (−0.882 + 2.86i)12-s + (−1.14 + 0.914i)13-s + (−3.10 − 3.88i)16-s + (−1.30 + 1.51i)18-s + (0.974 − 0.222i)23-s + (−1.03 − 3.85i)24-s + (−0.900 − 0.433i)25-s + (0.967 − 2.76i)26-s + (−0.900 + 0.433i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 - 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 - 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2164402762\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2164402762\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.988 - 0.149i)T \) |
| 23 | \( 1 + (-0.974 + 0.222i)T \) |
| 29 | \( 1 + (0.563 + 0.826i)T \) |
good | 2 | \( 1 + (1.69 - 1.06i)T + (0.433 - 0.900i)T^{2} \) |
| 5 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 7 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 11 | \( 1 + (0.974 + 0.222i)T^{2} \) |
| 13 | \( 1 + (1.14 - 0.914i)T + (0.222 - 0.974i)T^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + (-0.781 + 0.623i)T^{2} \) |
| 31 | \( 1 + (-1.02 - 1.63i)T + (-0.433 + 0.900i)T^{2} \) |
| 37 | \( 1 + (-0.974 + 0.222i)T^{2} \) |
| 41 | \( 1 + (0.565 - 0.565i)T - iT^{2} \) |
| 43 | \( 1 + (-0.433 - 0.900i)T^{2} \) |
| 47 | \( 1 + (0.369 + 0.0416i)T + (0.974 + 0.222i)T^{2} \) |
| 53 | \( 1 + (-0.900 - 0.433i)T^{2} \) |
| 59 | \( 1 - 1.80iT - T^{2} \) |
| 61 | \( 1 + (0.781 + 0.623i)T^{2} \) |
| 67 | \( 1 + (-0.222 - 0.974i)T^{2} \) |
| 71 | \( 1 + (-0.367 - 0.460i)T + (-0.222 + 0.974i)T^{2} \) |
| 73 | \( 1 + (0.806 - 1.28i)T + (-0.433 - 0.900i)T^{2} \) |
| 79 | \( 1 + (0.974 - 0.222i)T^{2} \) |
| 83 | \( 1 + (0.623 + 0.781i)T^{2} \) |
| 89 | \( 1 + (-0.433 + 0.900i)T^{2} \) |
| 97 | \( 1 + (0.781 - 0.623i)T^{2} \) |
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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
−9.708504015032349667060713406820, −8.953178828848571389316941626675, −8.133747827860928452696829063839, −7.16199174573195953465760086309, −6.86358623346853727540105508349, −6.03787540786941866114154661344, −5.22738225717061837306654565234, −4.52437411788101704768510644434, −2.36796812779140371690488613519, −1.19991699225866404565140561636,
0.34728250528033734823058044462, 1.61786924811916406230213698516, 2.64695073738592227748773922737, 3.69517669325971802560484180572, 4.85136164817021656938782752080, 6.05976318549451767531801125051, 7.08971638363295089464546542278, 7.55412132327584438993029463980, 8.237305172015228004653160055933, 9.382214201885955562093812823350