L(s) = 1 | + (−0.691 + 0.722i)2-s + (0.936 − 0.351i)3-s + (−0.0448 − 0.998i)4-s + (−0.809 + 0.587i)5-s + (−0.393 + 0.919i)6-s + (−0.134 − 0.990i)7-s + (0.753 + 0.657i)8-s + (0.753 − 0.657i)9-s + (0.134 − 0.990i)10-s + (−0.858 − 0.512i)11-s + (−0.393 − 0.919i)12-s + (−0.858 + 0.512i)13-s + (0.809 + 0.587i)14-s + (−0.550 + 0.834i)15-s + (−0.995 + 0.0896i)16-s + (−0.309 − 0.951i)17-s + ⋯ |
L(s) = 1 | + (−0.691 + 0.722i)2-s + (0.936 − 0.351i)3-s + (−0.0448 − 0.998i)4-s + (−0.809 + 0.587i)5-s + (−0.393 + 0.919i)6-s + (−0.134 − 0.990i)7-s + (0.753 + 0.657i)8-s + (0.753 − 0.657i)9-s + (0.134 − 0.990i)10-s + (−0.858 − 0.512i)11-s + (−0.393 − 0.919i)12-s + (−0.858 + 0.512i)13-s + (0.809 + 0.587i)14-s + (−0.550 + 0.834i)15-s + (−0.995 + 0.0896i)16-s + (−0.309 − 0.951i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0279 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0279 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5447315995 - 0.5601778326i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5447315995 - 0.5601778326i\) |
\(L(1)\) |
\(\approx\) |
\(0.7481058003 - 0.06416284189i\) |
\(L(1)\) |
\(\approx\) |
\(0.7481058003 - 0.06416284189i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 71 | \( 1 \) |
good | 2 | \( 1 + (-0.691 + 0.722i)T \) |
| 3 | \( 1 + (0.936 - 0.351i)T \) |
| 5 | \( 1 + (-0.809 + 0.587i)T \) |
| 7 | \( 1 + (-0.134 - 0.990i)T \) |
| 11 | \( 1 + (-0.858 - 0.512i)T \) |
| 13 | \( 1 + (-0.858 + 0.512i)T \) |
| 17 | \( 1 + (-0.309 - 0.951i)T \) |
| 19 | \( 1 + (-0.550 - 0.834i)T \) |
| 23 | \( 1 + (0.900 - 0.433i)T \) |
| 29 | \( 1 + (-0.963 - 0.266i)T \) |
| 31 | \( 1 + (0.995 + 0.0896i)T \) |
| 37 | \( 1 + (-0.900 - 0.433i)T \) |
| 41 | \( 1 + (0.222 + 0.974i)T \) |
| 43 | \( 1 + (0.983 - 0.178i)T \) |
| 47 | \( 1 + (-0.936 - 0.351i)T \) |
| 53 | \( 1 + (0.0448 - 0.998i)T \) |
| 59 | \( 1 + (0.393 + 0.919i)T \) |
| 61 | \( 1 + (-0.134 + 0.990i)T \) |
| 67 | \( 1 + (0.0448 + 0.998i)T \) |
| 73 | \( 1 + (-0.691 + 0.722i)T \) |
| 79 | \( 1 + (0.753 + 0.657i)T \) |
| 83 | \( 1 + (-0.393 - 0.919i)T \) |
| 89 | \( 1 + (-0.0448 + 0.998i)T \) |
| 97 | \( 1 + (0.222 - 0.974i)T \) |
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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
−31.334486826256180777906215789707, −30.997636759658696281467391203252, −29.434906344150716490682173149097, −28.14420413849524611304012551391, −27.540600892979506408869626899388, −26.42659941688220895013101299727, −25.429144855619855321445465642465, −24.4100818243288893550353207127, −22.58654427767292143146039256854, −21.30020844753810277740662025009, −20.55613494882580159668408771700, −19.421358794636754816023014529, −18.83080325429181185504202046128, −17.22611943805769314623802799678, −15.80036467401144282800367611569, −15.04541072849591215665954442045, −12.95156708932447208563953022709, −12.33751950090778536046691264838, −10.681797676650699863041463133874, −9.4509163538910224670398005837, −8.41961468666140089606580942525, −7.60309254604391605517398309278, −4.814303294766704410172769844888, −3.361088064345328452403322965742, −2.04206599989564652426899806946,
0.42613568408587043024800450937, 2.68630463040575195042504054764, 4.50590870323437545194926130664, 6.87686647410489039146934379081, 7.413750059242400085386437119045, 8.606255786954343378147270845642, 9.97919210083853916376281982934, 11.208070265253439583314158152371, 13.2697626346766305411380966387, 14.32668594569577505215143629234, 15.30351069610158334913188863741, 16.38000220670173353649343413527, 17.85039203047394144008512983071, 19.05848069100349203045555221526, 19.55777602445639608902016493498, 20.78718804835901568941903805826, 22.79718272426668044668576188130, 23.83034459886451121257062336580, 24.57004503145385628937058304696, 26.16477175358370971814862438403, 26.47598525313702653203083152072, 27.33827358963012531297270697285, 29.06658590635798068899023605792, 30.0342249238591229440210787304, 31.41710184619273682560797247809