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 | \( 1 + (-0.5 - 0.866i)T \) |
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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
−22.15454624269161366580663435187, −20.701928922289635552390259386215, −20.16595196120006812722142552893, −19.351611892953011563493657178675, −18.7505296501933203309997610869, −18.06849357605460337807611405899, −17.206542091790983591523217568896, −16.0058203628003105664626360095, −15.63914649782441009260633062366, −14.85345239538599292296713605862, −14.1714314890850750793637843657, −13.39888654736713887594234771432, −12.710705030444391646728579511083, −10.952366491454007246453409943074, −10.51780035723893277663090102271, −9.62560742948866689489852615196, −8.83324879598233795093343568970, −7.868991629459910785608201150939, −7.18571983039049350983598113819, −6.829791639410111766863116492645, −5.54337807280782587454623205379, −4.137926361286499205081362731351, −3.68330085857688876695659488398, −2.39697516116349669573865503785, −1.145545225057148710175778035130,
0.72199491132140321723337923872, 1.7993166673430103460815846984, 3.02543030446116232331441628061, 3.35961437972062399956356609965, 4.534829212215217230154148209875, 5.492640229963999633838625601758, 7.036065015304625351490380889866, 8.005635794857326296425002843903, 8.70194313110743535508206133087, 9.09043049758795972142599547027, 9.88743720823488643851382125911, 11.10905890889562539272559949125, 11.76580794668900549432118046498, 12.75279193031547909636197508916, 13.29412413326781679402374624823, 13.90175597478825371855176539201, 15.36657129392353156289528837563, 15.97447169826756324468078606976, 16.48467752077884604276502856029, 17.8621035029502554715509258048, 18.65179084545895075442380239183, 19.07620377036106747168709970548, 19.844751640162877142367991390158, 20.75089814618216075528001489925, 20.908547909330464969983663201225