L(s) = 1 | + (−0.913 − 0.406i)2-s + (0.669 − 0.743i)3-s + (0.669 + 0.743i)4-s + (−0.913 + 0.406i)6-s + (0.309 − 0.951i)7-s + (−0.309 − 0.951i)8-s + (−0.104 − 0.994i)9-s + 12-s + (0.104 + 0.994i)13-s + (−0.669 + 0.743i)14-s + (−0.104 + 0.994i)16-s + (−0.104 + 0.994i)17-s + (−0.309 + 0.951i)18-s + (−0.5 − 0.866i)21-s + (0.5 − 0.866i)23-s + (−0.913 − 0.406i)24-s + ⋯ |
L(s) = 1 | + (−0.913 − 0.406i)2-s + (0.669 − 0.743i)3-s + (0.669 + 0.743i)4-s + (−0.913 + 0.406i)6-s + (0.309 − 0.951i)7-s + (−0.309 − 0.951i)8-s + (−0.104 − 0.994i)9-s + 12-s + (0.104 + 0.994i)13-s + (−0.669 + 0.743i)14-s + (−0.104 + 0.994i)16-s + (−0.104 + 0.994i)17-s + (−0.309 + 0.951i)18-s + (−0.5 − 0.866i)21-s + (0.5 − 0.866i)23-s + (−0.913 − 0.406i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.392 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.392 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7012168593 - 1.062069090i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7012168593 - 1.062069090i\) |
\(L(1)\) |
\(\approx\) |
\(0.8115533159 - 0.4905264319i\) |
\(L(1)\) |
\(\approx\) |
\(0.8115533159 - 0.4905264319i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (-0.913 - 0.406i)T \) |
| 3 | \( 1 + (0.669 - 0.743i)T \) |
| 7 | \( 1 + (0.309 - 0.951i)T \) |
| 13 | \( 1 + (0.104 + 0.994i)T \) |
| 17 | \( 1 + (-0.104 + 0.994i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
| 29 | \( 1 + (0.669 + 0.743i)T \) |
| 31 | \( 1 + (0.809 - 0.587i)T \) |
| 37 | \( 1 + (0.309 - 0.951i)T \) |
| 41 | \( 1 + (0.669 - 0.743i)T \) |
| 43 | \( 1 + (-0.5 - 0.866i)T \) |
| 47 | \( 1 + (0.978 + 0.207i)T \) |
| 53 | \( 1 + (-0.104 - 0.994i)T \) |
| 59 | \( 1 + (0.978 - 0.207i)T \) |
| 61 | \( 1 + (-0.913 + 0.406i)T \) |
| 67 | \( 1 + (-0.5 + 0.866i)T \) |
| 71 | \( 1 + (0.104 - 0.994i)T \) |
| 73 | \( 1 + (-0.978 + 0.207i)T \) |
| 79 | \( 1 + (0.913 + 0.406i)T \) |
| 83 | \( 1 + (-0.809 - 0.587i)T \) |
| 89 | \( 1 + (0.5 - 0.866i)T \) |
| 97 | \( 1 + (0.913 + 0.406i)T \) |
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
−21.512362879558094773419130603658, −20.908168331523372020841888522820, −20.121834227076657364541293394911, −19.45809347318028055736119740510, −18.645765010015731621973292186177, −17.91453010049827050863117630542, −17.13866295522290566084566653806, −16.11684648689927001514802017090, −15.50725481381241775355334226743, −15.1007031867560786798968892615, −14.19805437814896840200930632028, −13.32911857999329200024961353319, −11.96495663480894099869204239880, −11.23675419482234069726255770997, −10.32527004119811576435387948259, −9.58791691845756433390982629271, −8.934817518604480869740474309938, −8.146931163061098533153069955262, −7.573160487054793799793072019243, −6.28423571023794242767790799509, −5.35688712657833724635404242580, −4.67756339217391788602690219486, −3.01823416029841913807657484882, −2.5613873948244552028889892537, −1.22746575912498939919536550797,
0.73938684937716177818304589520, 1.66176142513184237025672186072, 2.475327870862908782108073070302, 3.62487289823886044344500385219, 4.341559691360704819446339255119, 6.19406459121135731964674924182, 6.942574916909427162088496187609, 7.56401876673254894847822959341, 8.499345362837008880766083593325, 8.9880447376716245812046975204, 10.0849369536476748356604649070, 10.80466605568385786961669747992, 11.71022057049618417423650862599, 12.53013113325061320767507526269, 13.28369913023788561279359190212, 14.15215875251346773723710926890, 14.90867362479265876683989911407, 16.02905664669587857584450415673, 16.90727244644199081881437504234, 17.50592860567168392363861972186, 18.27669377906299068000235902155, 19.16926081525623644823279419184, 19.49557019467384634811430279402, 20.44529534774969080054348764411, 20.911643227081516409333021927373