L(s) = 1 | + (−0.5 + 0.866i)2-s + 3-s + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.5 + 0.866i)6-s + (−0.5 − 0.866i)7-s + 8-s + 9-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)12-s + 13-s + 14-s + (−0.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)2-s + 3-s + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.5 + 0.866i)6-s + (−0.5 − 0.866i)7-s + 8-s + 9-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)12-s + 13-s + 14-s + (−0.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.245 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.245 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8398666410 + 1.078843466i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8398666410 + 1.078843466i\) |
\(L(1)\) |
\(\approx\) |
\(0.8962950056 + 0.5151133994i\) |
\(L(1)\) |
\(\approx\) |
\(0.8962950056 + 0.5151133994i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 1009 | \( 1 \) |
good | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + (-0.5 + 0.866i)T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + (-0.5 - 0.866i)T \) |
| 41 | \( 1 + (-0.5 - 0.866i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + (-0.5 + 0.866i)T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + (-0.5 + 0.866i)T \) |
| 71 | \( 1 + (-0.5 + 0.866i)T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 + (-0.5 + 0.866i)T \) |
| 89 | \( 1 + (-0.5 + 0.866i)T \) |
| 97 | \( 1 + (-0.5 + 0.866i)T \) |
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
−20.908547909330464969983663201225, −20.75089814618216075528001489925, −19.844751640162877142367991390158, −19.07620377036106747168709970548, −18.65179084545895075442380239183, −17.8621035029502554715509258048, −16.48467752077884604276502856029, −15.97447169826756324468078606976, −15.36657129392353156289528837563, −13.90175597478825371855176539201, −13.29412413326781679402374624823, −12.75279193031547909636197508916, −11.76580794668900549432118046498, −11.10905890889562539272559949125, −9.88743720823488643851382125911, −9.09043049758795972142599547027, −8.70194313110743535508206133087, −8.005635794857326296425002843903, −7.036065015304625351490380889866, −5.492640229963999633838625601758, −4.534829212215217230154148209875, −3.35961437972062399956356609965, −3.02543030446116232331441628061, −1.7993166673430103460815846984, −0.72199491132140321723337923872,
1.145545225057148710175778035130, 2.39697516116349669573865503785, 3.68330085857688876695659488398, 4.137926361286499205081362731351, 5.54337807280782587454623205379, 6.829791639410111766863116492645, 7.18571983039049350983598113819, 7.868991629459910785608201150939, 8.83324879598233795093343568970, 9.62560742948866689489852615196, 10.51780035723893277663090102271, 10.952366491454007246453409943074, 12.710705030444391646728579511083, 13.39888654736713887594234771432, 14.1714314890850750793637843657, 14.85345239538599292296713605862, 15.63914649782441009260633062366, 16.0058203628003105664626360095, 17.206542091790983591523217568896, 18.06849357605460337807611405899, 18.7505296501933203309997610869, 19.351611892953011563493657178675, 20.16595196120006812722142552893, 20.701928922289635552390259386215, 22.15454624269161366580663435187