L(s) = 1 | + (−0.642 − 0.766i)2-s + (−0.893 − 0.448i)3-s + (−0.173 + 0.984i)4-s + (0.918 − 0.396i)5-s + (0.230 + 0.973i)6-s + (−0.993 − 0.116i)7-s + (0.866 − 0.5i)8-s + (0.597 + 0.802i)9-s + (−0.893 − 0.448i)10-s + (0.893 − 0.448i)11-s + (0.597 − 0.802i)12-s + (0.116 − 0.993i)13-s + (0.549 + 0.835i)14-s + (−0.998 − 0.0581i)15-s + (−0.939 − 0.342i)16-s + (0.342 − 0.939i)17-s + ⋯ |
L(s) = 1 | + (−0.642 − 0.766i)2-s + (−0.893 − 0.448i)3-s + (−0.173 + 0.984i)4-s + (0.918 − 0.396i)5-s + (0.230 + 0.973i)6-s + (−0.993 − 0.116i)7-s + (0.866 − 0.5i)8-s + (0.597 + 0.802i)9-s + (−0.893 − 0.448i)10-s + (0.893 − 0.448i)11-s + (0.597 − 0.802i)12-s + (0.116 − 0.993i)13-s + (0.549 + 0.835i)14-s + (−0.998 − 0.0581i)15-s + (−0.939 − 0.342i)16-s + (0.342 − 0.939i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0513 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0513 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.3672473244 - 0.3488490150i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.3672473244 - 0.3488490150i\) |
\(L(1)\) |
\(\approx\) |
\(0.4675246717 - 0.4108707524i\) |
\(L(1)\) |
\(\approx\) |
\(0.4675246717 - 0.4108707524i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 + (-0.642 - 0.766i)T \) |
| 3 | \( 1 + (-0.893 - 0.448i)T \) |
| 5 | \( 1 + (0.918 - 0.396i)T \) |
| 7 | \( 1 + (-0.993 - 0.116i)T \) |
| 11 | \( 1 + (0.893 - 0.448i)T \) |
| 13 | \( 1 + (0.116 - 0.993i)T \) |
| 17 | \( 1 + (0.342 - 0.939i)T \) |
| 19 | \( 1 + (-0.984 + 0.173i)T \) |
| 23 | \( 1 + (0.984 + 0.173i)T \) |
| 29 | \( 1 + (0.998 - 0.0581i)T \) |
| 31 | \( 1 + (-0.116 - 0.993i)T \) |
| 41 | \( 1 + (-0.5 + 0.866i)T \) |
| 43 | \( 1 + (-0.642 - 0.766i)T \) |
| 47 | \( 1 + (0.686 - 0.727i)T \) |
| 53 | \( 1 + (-0.396 - 0.918i)T \) |
| 59 | \( 1 + (-0.998 - 0.0581i)T \) |
| 61 | \( 1 + (0.727 - 0.686i)T \) |
| 67 | \( 1 + (-0.993 + 0.116i)T \) |
| 71 | \( 1 + (0.766 + 0.642i)T \) |
| 73 | \( 1 + (-0.893 + 0.448i)T \) |
| 79 | \( 1 + (-0.802 + 0.597i)T \) |
| 83 | \( 1 + (-0.835 + 0.549i)T \) |
| 89 | \( 1 + (0.230 + 0.973i)T \) |
| 97 | \( 1 + (-0.802 - 0.597i)T \) |
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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
−18.80987155841123277879505752389, −17.86355933336237076504751776168, −17.2474386504406298218948669188, −16.92324570881960135611481112443, −16.32241200889567025854053970902, −15.5446465240138000436455459514, −14.85412669010285917781465018833, −14.33423650098769277954580450070, −13.44209761737182150040687926417, −12.64302578511974547530998288931, −11.92141558616132805121425085449, −10.826305730064975343839535876489, −10.46068902212784758510753513382, −9.80856332961283712171619824559, −9.071026612387958251600228891000, −8.81297906788398312921527495165, −7.2492845676908121438527348586, −6.615124190451622814080932839400, −6.375502208121637276757350228540, −5.7299670267707245070914419001, −4.76261907189333091131233066339, −4.12211428377818544871011123942, −2.98471169877159442640834224036, −1.7302654436589805542506589502, −1.161170681661045486091190678851,
0.14268939944611561308075786200, 0.77800695401671572972434980016, 1.40264305288238995916493994833, 2.4328690857531904655063570027, 3.133476936125938946257977737241, 4.128641853073264204744764067205, 5.05447674875319732634425690174, 5.840102433318605826700183552454, 6.603967161250127927570611835000, 7.099284784054456234704129024281, 8.21305627408400506614189760429, 8.84312149051969287697959477154, 9.78845262599380569124931052555, 10.03584413291844144487735059162, 10.883348948628973799020823473677, 11.54038446345940709328295413900, 12.37422469946984043748784663315, 12.78803166663678240497990807092, 13.42780742744989457138652175780, 13.91094810237436941999017794976, 15.26172007419342869759942385683, 16.23751708932232497020170048295, 16.70733474023932641591962868231, 17.1810811422245448049663032764, 17.70434878051796208621869719048