L(s) = 1 | + (0.983 + 0.178i)2-s + (−0.691 + 0.722i)3-s + (0.936 + 0.351i)4-s + (−0.393 − 0.919i)5-s + (−0.809 + 0.587i)6-s + (0.309 + 0.951i)7-s + (0.858 + 0.512i)8-s + (−0.0448 − 0.998i)9-s + (−0.222 − 0.974i)10-s + (−0.900 + 0.433i)12-s + (0.753 − 0.657i)13-s + (0.134 + 0.990i)14-s + (0.936 + 0.351i)15-s + (0.753 + 0.657i)16-s + (0.753 + 0.657i)17-s + (0.134 − 0.990i)18-s + ⋯ |
L(s) = 1 | + (0.983 + 0.178i)2-s + (−0.691 + 0.722i)3-s + (0.936 + 0.351i)4-s + (−0.393 − 0.919i)5-s + (−0.809 + 0.587i)6-s + (0.309 + 0.951i)7-s + (0.858 + 0.512i)8-s + (−0.0448 − 0.998i)9-s + (−0.222 − 0.974i)10-s + (−0.900 + 0.433i)12-s + (0.753 − 0.657i)13-s + (0.134 + 0.990i)14-s + (0.936 + 0.351i)15-s + (0.753 + 0.657i)16-s + (0.753 + 0.657i)17-s + (0.134 − 0.990i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 473 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.596 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 473 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.596 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.844746266 + 0.9275287702i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.844746266 + 0.9275287702i\) |
\(L(1)\) |
\(\approx\) |
\(1.524821250 + 0.4663040324i\) |
\(L(1)\) |
\(\approx\) |
\(1.524821250 + 0.4663040324i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 43 | \( 1 \) |
good | 2 | \( 1 + (0.983 + 0.178i)T \) |
| 3 | \( 1 + (-0.691 + 0.722i)T \) |
| 5 | \( 1 + (-0.393 - 0.919i)T \) |
| 7 | \( 1 + (0.309 + 0.951i)T \) |
| 13 | \( 1 + (0.753 - 0.657i)T \) |
| 17 | \( 1 + (0.753 + 0.657i)T \) |
| 19 | \( 1 + (-0.550 - 0.834i)T \) |
| 23 | \( 1 + (0.623 + 0.781i)T \) |
| 29 | \( 1 + (-0.691 - 0.722i)T \) |
| 31 | \( 1 + (0.983 + 0.178i)T \) |
| 37 | \( 1 + (0.309 + 0.951i)T \) |
| 41 | \( 1 + (0.134 + 0.990i)T \) |
| 47 | \( 1 + (-0.550 - 0.834i)T \) |
| 53 | \( 1 + (-0.393 + 0.919i)T \) |
| 59 | \( 1 + (0.858 - 0.512i)T \) |
| 61 | \( 1 + (0.983 - 0.178i)T \) |
| 67 | \( 1 + (0.623 - 0.781i)T \) |
| 71 | \( 1 + (-0.0448 + 0.998i)T \) |
| 73 | \( 1 + (-0.995 + 0.0896i)T \) |
| 79 | \( 1 + (-0.809 - 0.587i)T \) |
| 83 | \( 1 + (0.983 - 0.178i)T \) |
| 89 | \( 1 + (-0.900 - 0.433i)T \) |
| 97 | \( 1 + (-0.0448 - 0.998i)T \) |
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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
−23.4905620608241975135644325869, −22.92498517640134910483276507276, −22.43146478361872105743133634882, −21.20319183590091083949677975438, −20.52446455022430169047031625163, −19.232719811930498623011306921290, −18.87474712700523682515885091383, −17.77449438987425723013277019251, −16.59912599273493368357298813701, −16.08982871352459509657470212922, −14.63282586550665610116208521874, −14.1680357740110493054200074223, −13.28450554926617379334308362507, −12.34085313878374441798357989097, −11.41672131006726926312393652496, −10.9053688251161012640735928881, −10.12889529002909100670536723614, −8.075775876000042854743473472290, −7.1696347273308840754406400988, −6.632890546096778639123906985413, −5.650859742673537216803708786784, −4.44982659848703518624944823293, −3.56523912656921286732284753055, −2.29100674098078336044159151430, −1.09642324365074935673225072987,
1.33700322206283863813384707770, 3.01036943098134535997591065036, 4.02685526502605552915323649350, 4.958116199848212862583594115061, 5.57755237415341322098472217303, 6.402100759557524499791263906565, 7.94025686744419352129757349738, 8.73250723231028635529519662800, 9.97217387393177697457394081661, 11.254468199916982664871455038690, 11.6632232665011933289658261916, 12.6483606150769429200653973290, 13.266666181751645743546850075864, 14.80986267751288709650940325603, 15.38840991566893070015176090463, 15.92422572271637772177853017425, 16.94056517795297324496964095416, 17.53274395292768932310454528055, 18.96278075663572688362454563112, 20.09110306113911062350434079276, 20.9832300694365295249250592341, 21.384769614144779262958145971385, 22.230687744623163789503150332394, 23.28326906539414030815380473889, 23.61319494719324306695376166529