L(s) = 1 | + (−0.665 + 0.746i)2-s + (−0.543 + 0.839i)3-s + (−0.114 − 0.993i)4-s + (0.720 − 0.693i)5-s + (−0.264 − 0.964i)6-s + (0.896 − 0.443i)7-s + (0.817 + 0.575i)8-s + (−0.409 − 0.912i)9-s + (0.0383 + 0.999i)10-s + (0.190 − 0.981i)11-s + (0.896 + 0.443i)12-s + (0.477 + 0.878i)13-s + (−0.264 + 0.964i)14-s + (0.190 + 0.981i)15-s + (−0.973 + 0.227i)16-s + (−0.997 + 0.0765i)17-s + ⋯ |
L(s) = 1 | + (−0.665 + 0.746i)2-s + (−0.543 + 0.839i)3-s + (−0.114 − 0.993i)4-s + (0.720 − 0.693i)5-s + (−0.264 − 0.964i)6-s + (0.896 − 0.443i)7-s + (0.817 + 0.575i)8-s + (−0.409 − 0.912i)9-s + (0.0383 + 0.999i)10-s + (0.190 − 0.981i)11-s + (0.896 + 0.443i)12-s + (0.477 + 0.878i)13-s + (−0.264 + 0.964i)14-s + (0.190 + 0.981i)15-s + (−0.973 + 0.227i)16-s + (−0.997 + 0.0765i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.742 + 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.742 + 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6673910357 + 0.2566513473i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6673910357 + 0.2566513473i\) |
\(L(1)\) |
\(\approx\) |
\(0.7329783633 + 0.2488519209i\) |
\(L(1)\) |
\(\approx\) |
\(0.7329783633 + 0.2488519209i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 83 | \( 1 \) |
good | 2 | \( 1 + (-0.665 + 0.746i)T \) |
| 3 | \( 1 + (-0.543 + 0.839i)T \) |
| 5 | \( 1 + (0.720 - 0.693i)T \) |
| 7 | \( 1 + (0.896 - 0.443i)T \) |
| 11 | \( 1 + (0.190 - 0.981i)T \) |
| 13 | \( 1 + (0.477 + 0.878i)T \) |
| 17 | \( 1 + (-0.997 + 0.0765i)T \) |
| 19 | \( 1 + (0.338 + 0.941i)T \) |
| 23 | \( 1 + (0.953 - 0.301i)T \) |
| 29 | \( 1 + (-0.771 + 0.636i)T \) |
| 31 | \( 1 + (0.817 - 0.575i)T \) |
| 37 | \( 1 + (-0.409 + 0.912i)T \) |
| 41 | \( 1 + (-0.665 - 0.746i)T \) |
| 43 | \( 1 + (0.988 - 0.152i)T \) |
| 47 | \( 1 + (-0.859 + 0.511i)T \) |
| 53 | \( 1 + (-0.859 - 0.511i)T \) |
| 59 | \( 1 + (-0.927 + 0.373i)T \) |
| 61 | \( 1 + (0.606 + 0.795i)T \) |
| 67 | \( 1 + (-0.973 + 0.227i)T \) |
| 71 | \( 1 + (0.896 + 0.443i)T \) |
| 73 | \( 1 + (0.0383 + 0.999i)T \) |
| 79 | \( 1 + (-0.114 - 0.993i)T \) |
| 89 | \( 1 + (-0.264 - 0.964i)T \) |
| 97 | \( 1 + (-0.264 + 0.964i)T \) |
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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
−30.47751188007216593009376496280, −29.76513690823174296031597589117, −28.5650352972724519520366442513, −27.98055708318998944519060954593, −26.631066035624637440273242570891, −25.376109091417055797136971610306, −24.69754225252032942916210865894, −22.93793280539652966973763078494, −22.14156076098164441538187383381, −20.984324481104305189901285623880, −19.75592450621442655144545577621, −18.50504509756328912631445879161, −17.71426661207792235746722767996, −17.37871573644577583089841814470, −15.34401608110144285059648220545, −13.69866140978908857674234531685, −12.726469190135928095323045369571, −11.41453507469144002004630440615, −10.7315553825123875128794454366, −9.22055609656304912237461364661, −7.79381384027718821599733514614, −6.6890246532680693607530785574, −5.00939723422504066555454472201, −2.6595592482315232926011879732, −1.586837187824997882876345420398,
1.33403682442014400372287433456, 4.34889166129586111728274103266, 5.4291344103581620662027070790, 6.52644092782139932293438403941, 8.44264857685545706176972746873, 9.23579954797484719524222967496, 10.538971527449402097997496901067, 11.46733087113469376183671837220, 13.61738422648723441030233396847, 14.58112272869170148606250758412, 15.990275320848712665235509662492, 16.82432685638755149779219493997, 17.46740156956440357038593321947, 18.70057572312741475516370755794, 20.409358315970117691554893351441, 21.16427840126755802070233425449, 22.49472413966809085061369336204, 23.91266621799351324197884843400, 24.446922877541217343560751197635, 25.90351035532011882256428296231, 26.87863804884221759487914577442, 27.604268500655831158457530115974, 28.72481893405441406695161888798, 29.35122062114101479855012665349, 31.36977616269933679079652192154