L(s) = 1 | + (0.623 + 0.781i)2-s + (−0.222 − 0.974i)3-s + (−0.222 + 0.974i)4-s + (0.623 + 0.781i)5-s + (0.623 − 0.781i)6-s + (−0.222 − 0.974i)7-s + (−0.900 + 0.433i)8-s + (−0.900 + 0.433i)9-s + (−0.222 + 0.974i)10-s + (−0.900 − 0.433i)11-s + 12-s + (−0.900 − 0.433i)13-s + (0.623 − 0.781i)14-s + (0.623 − 0.781i)15-s + (−0.900 − 0.433i)16-s + 17-s + ⋯ |
L(s) = 1 | + (0.623 + 0.781i)2-s + (−0.222 − 0.974i)3-s + (−0.222 + 0.974i)4-s + (0.623 + 0.781i)5-s + (0.623 − 0.781i)6-s + (−0.222 − 0.974i)7-s + (−0.900 + 0.433i)8-s + (−0.900 + 0.433i)9-s + (−0.222 + 0.974i)10-s + (−0.900 − 0.433i)11-s + 12-s + (−0.900 − 0.433i)13-s + (0.623 − 0.781i)14-s + (0.623 − 0.781i)15-s + (−0.900 − 0.433i)16-s + 17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.855 + 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.855 + 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8670025821 + 0.2419789471i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8670025821 + 0.2419789471i\) |
\(L(1)\) |
\(\approx\) |
\(1.107935304 + 0.2465575357i\) |
\(L(1)\) |
\(\approx\) |
\(1.107935304 + 0.2465575357i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 29 | \( 1 \) |
good | 2 | \( 1 + (0.623 + 0.781i)T \) |
| 3 | \( 1 + (-0.222 - 0.974i)T \) |
| 5 | \( 1 + (0.623 + 0.781i)T \) |
| 7 | \( 1 + (-0.222 - 0.974i)T \) |
| 11 | \( 1 + (-0.900 - 0.433i)T \) |
| 13 | \( 1 + (-0.900 - 0.433i)T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + (-0.222 + 0.974i)T \) |
| 23 | \( 1 + (0.623 - 0.781i)T \) |
| 31 | \( 1 + (0.623 + 0.781i)T \) |
| 37 | \( 1 + (-0.900 + 0.433i)T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 + (0.623 - 0.781i)T \) |
| 47 | \( 1 + (-0.900 - 0.433i)T \) |
| 53 | \( 1 + (0.623 + 0.781i)T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + (-0.222 - 0.974i)T \) |
| 67 | \( 1 + (-0.900 + 0.433i)T \) |
| 71 | \( 1 + (-0.900 - 0.433i)T \) |
| 73 | \( 1 + (0.623 - 0.781i)T \) |
| 79 | \( 1 + (-0.900 + 0.433i)T \) |
| 83 | \( 1 + (-0.222 + 0.974i)T \) |
| 89 | \( 1 + (0.623 + 0.781i)T \) |
| 97 | \( 1 + (-0.222 + 0.974i)T \) |
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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
−37.39027899654660773251304173249, −36.49248713724828751864705575186, −34.32700190206809129728835216859, −33.17066664678966439088990420063, −32.00319791325642011574843912590, −31.43322853231579063423886782330, −29.38177252567326692571952768931, −28.46364549863046286890333186446, −27.71427759350211912667125558644, −25.83854410020353207462493594906, −24.25221451545722138060684098177, −22.75725135996340718917740299621, −21.463178604150063307236876344128, −20.99954262032163585143748860372, −19.42733449376387309090769850350, −17.62570589298972592738392630703, −15.91086718365607048026202776731, −14.69076706307787309494151081602, −12.96277523568800026875007687731, −11.74763329073383371968176783727, −10.02645301479179505306856722472, −9.14032345328068345315784728183, −5.66282989807631649012267817691, −4.76049809736198473232714243615, −2.61651033357808942884242111086,
2.95339600789876898960738874757, 5.49183744252799304859137727700, 6.84017801857655669265959072418, 7.89168470411729630356460724305, 10.478710533679796669406020076019, 12.449968794676153735961084861045, 13.63032461782041315813390405205, 14.56242637685005727140846513990, 16.617693794563357237382129598645, 17.643664362639249722445539823171, 18.93799746367965548045941164598, 20.959154739538277713774750518522, 22.58576655421226264164138668789, 23.33703394391592055027966476857, 24.654817954722246583627492623241, 25.75999515109237803186809704530, 26.83677781008153409066475554693, 29.34434995150138709418098390670, 29.87422545017352444003071125672, 31.15216951447500548655986026265, 32.565642483692090955492166232507, 33.97439334764516360642997561299, 34.54052223603958905751855953794, 36.07784148589745953785555874544, 36.94961694925969965086927833969