L(s) = 1 | + (0.669 + 0.743i)2-s + (0.913 + 0.406i)3-s + (−0.104 + 0.994i)4-s + (−0.978 − 0.207i)5-s + (0.309 + 0.951i)6-s + (−0.809 + 0.587i)8-s + (0.669 + 0.743i)9-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)12-s + (0.309 − 0.951i)13-s + (−0.809 − 0.587i)15-s + (−0.978 − 0.207i)16-s + (0.669 − 0.743i)17-s + (−0.104 + 0.994i)18-s + (−0.104 − 0.994i)19-s + (0.309 − 0.951i)20-s + ⋯ |
L(s) = 1 | + (0.669 + 0.743i)2-s + (0.913 + 0.406i)3-s + (−0.104 + 0.994i)4-s + (−0.978 − 0.207i)5-s + (0.309 + 0.951i)6-s + (−0.809 + 0.587i)8-s + (0.669 + 0.743i)9-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)12-s + (0.309 − 0.951i)13-s + (−0.809 − 0.587i)15-s + (−0.978 − 0.207i)16-s + (0.669 − 0.743i)17-s + (−0.104 + 0.994i)18-s + (−0.104 − 0.994i)19-s + (0.309 − 0.951i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0870 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0870 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.077846772 + 0.9877809400i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.077846772 + 0.9877809400i\) |
\(L(1)\) |
\(\approx\) |
\(1.297672501 + 0.7790385360i\) |
\(L(1)\) |
\(\approx\) |
\(1.297672501 + 0.7790385360i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.669 + 0.743i)T \) |
| 3 | \( 1 + (0.913 + 0.406i)T \) |
| 5 | \( 1 + (-0.978 - 0.207i)T \) |
| 13 | \( 1 + (0.309 - 0.951i)T \) |
| 17 | \( 1 + (0.669 - 0.743i)T \) |
| 19 | \( 1 + (-0.104 - 0.994i)T \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
| 29 | \( 1 + (-0.809 - 0.587i)T \) |
| 31 | \( 1 + (-0.978 + 0.207i)T \) |
| 37 | \( 1 + (0.913 - 0.406i)T \) |
| 41 | \( 1 + (-0.809 + 0.587i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (-0.104 - 0.994i)T \) |
| 53 | \( 1 + (-0.978 + 0.207i)T \) |
| 59 | \( 1 + (-0.104 + 0.994i)T \) |
| 61 | \( 1 + (-0.978 - 0.207i)T \) |
| 67 | \( 1 + (-0.5 - 0.866i)T \) |
| 71 | \( 1 + (0.309 + 0.951i)T \) |
| 73 | \( 1 + (-0.104 + 0.994i)T \) |
| 79 | \( 1 + (0.669 + 0.743i)T \) |
| 83 | \( 1 + (0.309 + 0.951i)T \) |
| 89 | \( 1 + (-0.5 + 0.866i)T \) |
| 97 | \( 1 + (0.309 - 0.951i)T \) |
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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
−30.96776389807897477799050339202, −30.30660625033798915927631292054, −29.17589640226735494675897913518, −27.86858005659016004438590083136, −26.83276103672911397916842757374, −25.60955644085789686219251774722, −24.092785208774933087951198431969, −23.61375630475274514580503685340, −22.27023564981005092805027758795, −20.95237350028309256483508716156, −20.13521293333882290598786726675, −19.018797519589153132779611078500, −18.56110316881402305754620917904, −16.21661082359214147016086771515, −14.84755226907878755538677653623, −14.26216428819812681825326871651, −12.822538662864294086654898495178, −11.98040335166072241800330194065, −10.62782543526104499845491464518, −9.15494903348111520664321900691, −7.80262004878191632269846076261, −6.32440328551947295994073722629, −4.23041519628241824158797335031, −3.35077669443214817572836412607, −1.73665983393474109816296844380,
2.990833218132938635292311455936, 4.02306846432650952566367312208, 5.3100935802275824852196302679, 7.329053616237217400406805874116, 8.07903797534328757884524911726, 9.31262103810417548173007021334, 11.23139363422244992907978113385, 12.667652987561066263872993218632, 13.71155476539002874862799238174, 14.99149269834527046115497303196, 15.62712098900070918437734286691, 16.610941533829054281871391032615, 18.277219849377688680765484174112, 19.787527217514119985150389344490, 20.62040536756270822380014013154, 21.827754251923311386478910637480, 22.96548484887485251629940371728, 24.01854500385135383505823009013, 25.07355125292295264200875878384, 25.972045222160691265002976079322, 27.06201731281335474439170176930, 27.83054175733286694480871775958, 29.99619603267166464033858133474, 30.72860168909922521528719221633, 31.84905342887138728727433424263