L(s) = 1 | + (0.448 + 0.893i)2-s + (−0.597 + 0.802i)4-s + (0.918 + 0.396i)7-s + (−0.984 − 0.173i)8-s + (0.973 + 0.230i)11-s + (0.549 + 0.835i)13-s + (0.0581 + 0.998i)14-s + (−0.286 − 0.957i)16-s + (0.342 − 0.939i)17-s + (0.939 − 0.342i)19-s + (0.230 + 0.973i)22-s + (0.918 − 0.396i)23-s + (−0.5 + 0.866i)26-s + (−0.866 + 0.5i)28-s + (0.0581 − 0.998i)29-s + ⋯ |
L(s) = 1 | + (0.448 + 0.893i)2-s + (−0.597 + 0.802i)4-s + (0.918 + 0.396i)7-s + (−0.984 − 0.173i)8-s + (0.973 + 0.230i)11-s + (0.549 + 0.835i)13-s + (0.0581 + 0.998i)14-s + (−0.286 − 0.957i)16-s + (0.342 − 0.939i)17-s + (0.939 − 0.342i)19-s + (0.230 + 0.973i)22-s + (0.918 − 0.396i)23-s + (−0.5 + 0.866i)26-s + (−0.866 + 0.5i)28-s + (0.0581 − 0.998i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.222 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.222 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.903235012 + 2.385361490i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.903235012 + 2.385361490i\) |
\(L(1)\) |
\(\approx\) |
\(1.321585226 + 0.8918795715i\) |
\(L(1)\) |
\(\approx\) |
\(1.321585226 + 0.8918795715i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (0.448 + 0.893i)T \) |
| 7 | \( 1 + (0.918 + 0.396i)T \) |
| 11 | \( 1 + (0.973 + 0.230i)T \) |
| 13 | \( 1 + (0.549 + 0.835i)T \) |
| 17 | \( 1 + (0.342 - 0.939i)T \) |
| 19 | \( 1 + (0.939 - 0.342i)T \) |
| 23 | \( 1 + (0.918 - 0.396i)T \) |
| 29 | \( 1 + (0.0581 - 0.998i)T \) |
| 31 | \( 1 + (-0.993 - 0.116i)T \) |
| 37 | \( 1 + (0.642 + 0.766i)T \) |
| 41 | \( 1 + (0.893 + 0.448i)T \) |
| 43 | \( 1 + (0.727 + 0.686i)T \) |
| 47 | \( 1 + (-0.116 - 0.993i)T \) |
| 53 | \( 1 + (-0.866 + 0.5i)T \) |
| 59 | \( 1 + (-0.973 + 0.230i)T \) |
| 61 | \( 1 + (0.597 + 0.802i)T \) |
| 67 | \( 1 + (0.998 - 0.0581i)T \) |
| 71 | \( 1 + (0.173 + 0.984i)T \) |
| 73 | \( 1 + (-0.984 - 0.173i)T \) |
| 79 | \( 1 + (-0.893 + 0.448i)T \) |
| 83 | \( 1 + (-0.448 - 0.893i)T \) |
| 89 | \( 1 + (-0.173 + 0.984i)T \) |
| 97 | \( 1 + (0.957 - 0.286i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−23.7256997368704793127271644516, −22.9157664547310643337096951101, −22.05105114436908620429773730793, −21.241635975870125206654482174502, −20.423557716134823258524055673507, −19.78011073723569140674738550981, −18.79139260913617078928769949422, −17.86251764735805503095072669894, −17.142722723549748630617479868542, −15.796167219953215000072594270650, −14.55583568188087203568682104233, −14.26866098396181255854327654986, −13.067422328600059780644860723397, −12.29625809296963884989398512155, −11.13462563816737620076494769782, −10.79330066317575740457537683786, −9.5242239677856388592457947779, −8.61379441089391490983244351600, −7.47217906890890666010048912891, −6.000849138854382651638196896980, −5.17736821340652746368039436867, −3.983325556253383820865338334, −3.24527373132288084983876647234, −1.65296607103442462637869072932, −0.92799652665219963268219076857,
1.13368954687407811887427042868, 2.74405259841490453625342685906, 4.08846384034406110017309535214, 4.88231637820617773603907516005, 5.91229253791907638464248670565, 6.94054227861887027844088625362, 7.78392067303149900823715512610, 8.91084698264302721816338296328, 9.48937360873938660619203986090, 11.38527568531060446039342826802, 11.78551492553335542461032517408, 13.01609956761127254190337829597, 14.0758742401080749528406204442, 14.53867567348668580098993593297, 15.52253920867867399832657898327, 16.43411919556334346969318885875, 17.22360499601273444348262941722, 18.13936810125737522703692258561, 18.81843323730074198516218671417, 20.2881013602629235076987907464, 21.11726306277560775980800179844, 21.89959701665566896328585208392, 22.77229363894549120613295972805, 23.54687696674505643735045665588, 24.61351622877759719633700384512