L(s) = 1 | + (−0.991 − 0.130i)2-s + (0.793 − 0.608i)3-s + (0.965 + 0.258i)4-s + (0.896 − 0.442i)5-s + (−0.866 + 0.5i)6-s + (−0.0654 − 0.997i)7-s + (−0.923 − 0.382i)8-s + (0.258 − 0.965i)9-s + (−0.946 + 0.321i)10-s + (−0.130 − 0.991i)11-s + (0.923 − 0.382i)12-s + (0.442 + 0.896i)13-s + (−0.0654 + 0.997i)14-s + (0.442 − 0.896i)15-s + (0.866 + 0.5i)16-s + (−0.997 − 0.0654i)17-s + ⋯ |
L(s) = 1 | + (−0.991 − 0.130i)2-s + (0.793 − 0.608i)3-s + (0.965 + 0.258i)4-s + (0.896 − 0.442i)5-s + (−0.866 + 0.5i)6-s + (−0.0654 − 0.997i)7-s + (−0.923 − 0.382i)8-s + (0.258 − 0.965i)9-s + (−0.946 + 0.321i)10-s + (−0.130 − 0.991i)11-s + (0.923 − 0.382i)12-s + (0.442 + 0.896i)13-s + (−0.0654 + 0.997i)14-s + (0.442 − 0.896i)15-s + (0.866 + 0.5i)16-s + (−0.997 − 0.0654i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.312 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.312 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9293770458 - 1.284072908i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9293770458 - 1.284072908i\) |
\(L(1)\) |
\(\approx\) |
\(0.9290354986 - 0.5320882069i\) |
\(L(1)\) |
\(\approx\) |
\(0.9290354986 - 0.5320882069i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 97 | \( 1 \) |
good | 2 | \( 1 + (-0.991 - 0.130i)T \) |
| 3 | \( 1 + (0.793 - 0.608i)T \) |
| 5 | \( 1 + (0.896 - 0.442i)T \) |
| 7 | \( 1 + (-0.0654 - 0.997i)T \) |
| 11 | \( 1 + (-0.130 - 0.991i)T \) |
| 13 | \( 1 + (0.442 + 0.896i)T \) |
| 17 | \( 1 + (-0.997 - 0.0654i)T \) |
| 19 | \( 1 + (0.831 + 0.555i)T \) |
| 23 | \( 1 + (-0.751 + 0.659i)T \) |
| 29 | \( 1 + (0.946 + 0.321i)T \) |
| 31 | \( 1 + (-0.608 - 0.793i)T \) |
| 37 | \( 1 + (-0.659 + 0.751i)T \) |
| 41 | \( 1 + (0.321 - 0.946i)T \) |
| 43 | \( 1 + (-0.258 - 0.965i)T \) |
| 47 | \( 1 + (0.707 - 0.707i)T \) |
| 53 | \( 1 + (-0.130 + 0.991i)T \) |
| 59 | \( 1 + (0.751 + 0.659i)T \) |
| 61 | \( 1 + (-0.5 + 0.866i)T \) |
| 67 | \( 1 + (0.555 - 0.831i)T \) |
| 71 | \( 1 + (0.321 + 0.946i)T \) |
| 73 | \( 1 + (0.965 - 0.258i)T \) |
| 79 | \( 1 + (0.382 - 0.923i)T \) |
| 83 | \( 1 + (0.0654 - 0.997i)T \) |
| 89 | \( 1 + (0.923 + 0.382i)T \) |
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
−30.19118072269472008721832863042, −28.668349459222484281270933907017, −28.12781523512568182818587516711, −26.84972713481668962965770106937, −25.976778200880488964180874813626, −25.24919356906548940262403486873, −24.60471739017703318468498199306, −22.50778648769822557284975694709, −21.50788543038763151819758921441, −20.494205303710455982853376628157, −19.611123263875774348925926360279, −18.19674434690603622577345121301, −17.74125208999866456432212683038, −15.99180858678830658524625508047, −15.31397563500441152683262387312, −14.28733683611854945303160339335, −12.73523578550753749920919853045, −10.99012096238930163892178231555, −9.9546459832633597997038925167, −9.186552027114005441790384266173, −8.10445630132372670172860660459, −6.62466768278861398642814500232, −5.22132093572700323733679608974, −2.898659544267440755836282439522, −2.0110079306723282369830615838,
0.89899368491977522025938143715, 2.045163861027081637764963434062, 3.64373162049958775786820392050, 6.11859024786406633363707842548, 7.20343880186537649495774158795, 8.47674485017837761293774397056, 9.31868680091801231496199459033, 10.48349084303892247885500620753, 11.91830907757062622637676786662, 13.44623095774921770336698902043, 14.00467851376937291645750138009, 15.85701640190595942399574137098, 16.88849439958592678037271307513, 17.92841274866457608555948713886, 18.8549051856058725811667629938, 19.99936701810956358796870655414, 20.66757377940709171795708008857, 21.71442006785577744054885943021, 23.8975896027431228992188996382, 24.42479397155750715678369557231, 25.63861470206036145508706353340, 26.29178037756109820768115437584, 27.18349771420640982276423441143, 28.89162038308395250096442562636, 29.28621151290284966852641983403