L(s) = 1 | + 2-s + (0.5 + 0.866i)3-s + 4-s + (−0.228 − 0.395i)5-s + (0.5 + 0.866i)6-s + (−0.369 + 2.61i)7-s + 8-s + (−0.499 + 0.866i)9-s + (−0.228 − 0.395i)10-s + (1.91 + 3.32i)11-s + (0.5 + 0.866i)12-s + (−3.13 + 1.78i)13-s + (−0.369 + 2.61i)14-s + (0.228 − 0.395i)15-s + 16-s + 1.55·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (0.288 + 0.499i)3-s + 0.5·4-s + (−0.102 − 0.176i)5-s + (0.204 + 0.353i)6-s + (−0.139 + 0.990i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.0721 − 0.124i)10-s + (0.578 + 1.00i)11-s + (0.144 + 0.249i)12-s + (−0.869 + 0.494i)13-s + (−0.0988 + 0.700i)14-s + (0.0589 − 0.102i)15-s + 0.250·16-s + 0.376·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.460 - 0.887i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.460 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.99111 + 1.21036i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.99111 + 1.21036i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (0.369 - 2.61i)T \) |
| 13 | \( 1 + (3.13 - 1.78i)T \) |
good | 5 | \( 1 + (0.228 + 0.395i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.91 - 3.32i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 - 1.55T + 17T^{2} \) |
| 19 | \( 1 + (-1.44 + 2.49i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 3.24T + 23T^{2} \) |
| 29 | \( 1 + (2.20 - 3.82i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.80 + 8.31i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 0.280T + 37T^{2} \) |
| 41 | \( 1 + (3.57 - 6.18i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.21 - 2.10i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (3.93 + 6.80i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.550 + 0.953i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 9.36T + 59T^{2} \) |
| 61 | \( 1 + (-5.55 + 9.61i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.894 - 1.55i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (5.06 + 8.77i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.40 + 2.43i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.70 - 4.69i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 1.35T + 83T^{2} \) |
| 89 | \( 1 + 0.179T + 89T^{2} \) |
| 97 | \( 1 + (-4.73 - 8.20i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.17659071458166659190314621609, −9.853168595374057002966606545647, −9.407647364152835923790715966633, −8.344334818092892796801049057025, −7.22176572445644725005814393672, −6.29789140437920821954041407896, −5.04730950327137003606592713419, −4.50917239324284242106705905436, −3.14721144183630960173121111191, −2.09574131163699374491228031648,
1.16357820883724181801072898118, 2.96011895992169753206188018258, 3.69503428115492167533167241869, 5.00176765877089241059879115088, 6.10132811065521878119733904502, 7.06234514150626502589475117593, 7.66696845111927181349633700215, 8.754945931419918523226661850530, 9.945397086530662008837107303227, 10.76916855111524583241103288465