L(s) = 1 | + (0.278 + 0.960i)2-s + (−0.948 − 0.316i)3-s + (−0.845 + 0.534i)4-s + (0.428 + 0.903i)5-s + (0.0402 − 0.999i)6-s + (0.200 + 0.979i)7-s + (−0.748 − 0.663i)8-s + (0.799 + 0.600i)9-s + (−0.748 + 0.663i)10-s + (−0.996 + 0.0804i)11-s + (0.970 − 0.239i)12-s + (−0.0402 − 0.999i)13-s + (−0.885 + 0.464i)14-s + (−0.120 − 0.992i)15-s + (0.428 − 0.903i)16-s + (−0.885 − 0.464i)17-s + ⋯ |
L(s) = 1 | + (0.278 + 0.960i)2-s + (−0.948 − 0.316i)3-s + (−0.845 + 0.534i)4-s + (0.428 + 0.903i)5-s + (0.0402 − 0.999i)6-s + (0.200 + 0.979i)7-s + (−0.748 − 0.663i)8-s + (0.799 + 0.600i)9-s + (−0.748 + 0.663i)10-s + (−0.996 + 0.0804i)11-s + (0.970 − 0.239i)12-s + (−0.0402 − 0.999i)13-s + (−0.885 + 0.464i)14-s + (−0.120 − 0.992i)15-s + (0.428 − 0.903i)16-s + (−0.885 − 0.464i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.576 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.576 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.2087390366 + 0.4028048234i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.2087390366 + 0.4028048234i\) |
\(L(1)\) |
\(\approx\) |
\(0.5056604298 + 0.4667572946i\) |
\(L(1)\) |
\(\approx\) |
\(0.5056604298 + 0.4667572946i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 79 | \( 1 \) |
good | 2 | \( 1 + (0.278 + 0.960i)T \) |
| 3 | \( 1 + (-0.948 - 0.316i)T \) |
| 5 | \( 1 + (0.428 + 0.903i)T \) |
| 7 | \( 1 + (0.200 + 0.979i)T \) |
| 11 | \( 1 + (-0.996 + 0.0804i)T \) |
| 13 | \( 1 + (-0.0402 - 0.999i)T \) |
| 17 | \( 1 + (-0.885 - 0.464i)T \) |
| 19 | \( 1 + (-0.632 - 0.774i)T \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
| 29 | \( 1 + (0.919 + 0.391i)T \) |
| 31 | \( 1 + (0.692 - 0.721i)T \) |
| 37 | \( 1 + (-0.987 + 0.160i)T \) |
| 41 | \( 1 + (-0.568 + 0.822i)T \) |
| 43 | \( 1 + (0.996 + 0.0804i)T \) |
| 47 | \( 1 + (-0.987 - 0.160i)T \) |
| 53 | \( 1 + (-0.948 + 0.316i)T \) |
| 59 | \( 1 + (0.845 + 0.534i)T \) |
| 61 | \( 1 + (0.354 - 0.935i)T \) |
| 67 | \( 1 + (-0.970 + 0.239i)T \) |
| 71 | \( 1 + (0.748 + 0.663i)T \) |
| 73 | \( 1 + (-0.0402 + 0.999i)T \) |
| 83 | \( 1 + (-0.845 + 0.534i)T \) |
| 89 | \( 1 + (-0.748 + 0.663i)T \) |
| 97 | \( 1 + (-0.354 + 0.935i)T \) |
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
−29.87159167715859837180787619716, −28.88425247205193120877023101214, −28.52614106454640154568761494221, −27.2448581514344062405739422443, −26.34492291809314985463682712101, −24.13093813190475277935858093984, −23.64835116416968234700333958514, −22.487903602381370863971634595775, −21.13866799775846554473270728817, −20.85858138961286575479106735392, −19.37503035017835976787533417507, −17.936428201693665036428761314, −17.10015032990833728732127086981, −15.93414281000545144148071881444, −14.101970575251340518561599373090, −13.01471337648301313949776243857, −12.0639240181291894049857320569, −10.71809822946092188885438571058, −10.039100458884774024811797712987, −8.5370234537247067615325944454, −6.32173044877198126612459750853, −4.88837453439643980921624974655, −4.15479468543507171061583199743, −1.74991001607292053345525787866, −0.22055314663802217399080278818,
2.65686925934767699687626992885, 4.93507589894562404898949647228, 5.85578405402496335504431261914, 6.8911586959560736668336323116, 8.15725561638038807105757270538, 9.91537183403233179811599363938, 11.27788705717890500224639737882, 12.68804149053996315933433263300, 13.63470323206760375649650055713, 15.24899945692820702577956589142, 15.78594745773183833476967387396, 17.63305832348557818377620521934, 17.872718728291117289726018047, 18.94832777196384324221754975932, 21.352990831440575697645054340090, 22.12938152773048180732397696581, 22.93189149556275525485390514745, 24.021081780447222869211259616930, 25.03823356172052520639332816417, 25.977484634186315006376091597528, 27.242649874876449008823976434877, 28.25621010534622193413380522621, 29.54280755344933373241301552912, 30.57789145498471059464522415632, 31.58386292160122561862789639127