L(s) = 1 | + (0.137 − 0.238i)2-s + (−0.732 + 1.26i)3-s + (0.961 + 1.66i)4-s − 1.37·5-s + (0.202 + 0.350i)6-s + (0.239 + 0.415i)7-s + 1.08·8-s + (0.427 + 0.740i)9-s + (−0.189 + 0.328i)10-s + (0.323 − 0.561i)11-s − 2.81·12-s + (−3.42 + 1.12i)13-s + 0.132·14-s + (1.00 − 1.74i)15-s + (−1.77 + 3.07i)16-s + (1.17 + 2.03i)17-s + ⋯ |
L(s) = 1 | + (0.0975 − 0.168i)2-s + (−0.422 + 0.732i)3-s + (0.480 + 0.833i)4-s − 0.614·5-s + (0.0824 + 0.142i)6-s + (0.0906 + 0.156i)7-s + 0.382·8-s + (0.142 + 0.246i)9-s + (−0.0599 + 0.103i)10-s + (0.0976 − 0.169i)11-s − 0.813·12-s + (−0.949 + 0.312i)13-s + 0.0353·14-s + (0.259 − 0.450i)15-s + (−0.443 + 0.768i)16-s + (0.284 + 0.493i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.542 - 0.840i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.542 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.522868 + 0.960001i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.522868 + 0.960001i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 + (3.42 - 1.12i)T \) |
| 31 | \( 1 - T \) |
good | 2 | \( 1 + (-0.137 + 0.238i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (0.732 - 1.26i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + 1.37T + 5T^{2} \) |
| 7 | \( 1 + (-0.239 - 0.415i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.323 + 0.561i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.17 - 2.03i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.704 + 1.21i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.994 - 1.72i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.107 - 0.185i)T + (-14.5 - 25.1i)T^{2} \) |
| 37 | \( 1 + (3.54 - 6.14i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.27 + 9.13i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.55 - 7.88i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 8.99T + 47T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 59 | \( 1 + (-0.838 - 1.45i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.454 - 0.786i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.20 - 3.81i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.48 - 4.29i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 6.35T + 73T^{2} \) |
| 79 | \( 1 + 2.73T + 79T^{2} \) |
| 83 | \( 1 - 6.40T + 83T^{2} \) |
| 89 | \( 1 + (-2.40 + 4.15i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (5.40 + 9.36i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.64001491650545729359678870906, −10.78498587701972989274266808739, −9.988248300155668020590053289446, −8.795494841444827130518561825073, −7.77514552363518601753402666611, −7.09819474431407397066202060696, −5.68396386422635660678201939916, −4.47396333466932995801650985188, −3.74178984838032847100235760742, −2.29683951637032539130883731756,
0.71976912544673035618309089835, 2.27981251122676967643305612485, 4.08512994122774697501996812005, 5.34135346321677664066729295474, 6.24630069074887140790680758406, 7.25126177662127938114765019015, 7.67693904613013948911936213325, 9.258746573100143175956905082852, 10.16706889218614735399358290782, 11.04212703683183679238698231568