L(s) = 1 | + (−0.5 + 0.866i)3-s + (−0.669 + 0.743i)5-s + (−0.5 − 0.866i)9-s + (−0.669 − 0.743i)11-s + (0.809 + 0.587i)13-s + (−0.309 − 0.951i)15-s + (−0.669 − 0.743i)17-s + (0.913 + 0.406i)19-s + (0.104 − 0.994i)23-s + (−0.104 − 0.994i)25-s + 27-s + (0.309 + 0.951i)29-s + (0.669 + 0.743i)31-s + (0.978 − 0.207i)33-s + (0.669 − 0.743i)37-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)3-s + (−0.669 + 0.743i)5-s + (−0.5 − 0.866i)9-s + (−0.669 − 0.743i)11-s + (0.809 + 0.587i)13-s + (−0.309 − 0.951i)15-s + (−0.669 − 0.743i)17-s + (0.913 + 0.406i)19-s + (0.104 − 0.994i)23-s + (−0.104 − 0.994i)25-s + 27-s + (0.309 + 0.951i)29-s + (0.669 + 0.743i)31-s + (0.978 − 0.207i)33-s + (0.669 − 0.743i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.120 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.120 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7199412988 + 0.6379501676i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7199412988 + 0.6379501676i\) |
\(L(1)\) |
\(\approx\) |
\(0.7489339054 + 0.2932860549i\) |
\(L(1)\) |
\(\approx\) |
\(0.7489339054 + 0.2932860549i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 \) |
good | 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 + (-0.669 + 0.743i)T \) |
| 11 | \( 1 + (-0.669 - 0.743i)T \) |
| 13 | \( 1 + (0.809 + 0.587i)T \) |
| 17 | \( 1 + (-0.669 - 0.743i)T \) |
| 19 | \( 1 + (0.913 + 0.406i)T \) |
| 23 | \( 1 + (0.104 - 0.994i)T \) |
| 29 | \( 1 + (0.309 + 0.951i)T \) |
| 31 | \( 1 + (0.669 + 0.743i)T \) |
| 37 | \( 1 + (0.669 - 0.743i)T \) |
| 43 | \( 1 + (0.809 + 0.587i)T \) |
| 47 | \( 1 + (-0.104 + 0.994i)T \) |
| 53 | \( 1 + (-0.978 - 0.207i)T \) |
| 59 | \( 1 + (0.913 - 0.406i)T \) |
| 61 | \( 1 + (-0.913 - 0.406i)T \) |
| 67 | \( 1 + (0.978 + 0.207i)T \) |
| 71 | \( 1 + (-0.309 + 0.951i)T \) |
| 73 | \( 1 + (0.5 - 0.866i)T \) |
| 79 | \( 1 + (0.5 + 0.866i)T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + (-0.913 - 0.406i)T \) |
| 97 | \( 1 + (-0.309 - 0.951i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−20.92183661456018895238601317152, −20.25022933274990833658915651709, −19.5764291032443235490999069604, −18.82155578037810168768579037304, −17.90454911568785197257107224504, −17.439156695696029420829227242893, −16.56558451790582842257619848840, −15.56425564856249355966927413551, −15.28527194377285213863252702790, −13.65655303948121500496433992970, −13.25487407342818730273798233003, −12.5296533358872281464677228970, −11.71992697852957631869646834275, −11.13070922306478802610162542281, −10.11981869020835482491620767870, −9.01277308239534415802609689481, −7.98254527220561294134380068116, −7.70760376096357258618404704201, −6.60207654496512615120620894209, −5.662774210044929990382655124448, −4.92520757783257063608637255248, −3.9557140780911219157990532647, −2.70879983228180013125814540460, −1.58938350413911584833539964616, −0.61638298894318211647845738118,
0.833822333827061808545446480357, 2.68485688533096677626951221781, 3.35676038299048515988039018401, 4.27087059415754262984439266523, 5.09368626050261880813077959585, 6.15585030554817407230488121658, 6.79597753111837883940860946515, 7.93835832047134488067506482659, 8.77843921400232054522010166469, 9.65790969868399994724630580975, 10.734172998334787422044213891687, 11.029164631964837182535222748127, 11.77178964619689602704616679522, 12.71641011116597766287715447525, 14.03487834112903456593018310732, 14.38634779105157920519029175948, 15.5995626390167082500494128952, 16.0125411625338295303197765146, 16.46197854731950786024563003220, 17.81634728298168804492505913121, 18.28593819521171660602022010173, 19.06698612502341575969173091616, 20.07779377764962771126915033899, 20.80283660784058139279946551804, 21.51518478742727313207878200807