L(s) = 1 | + (−0.861 − 0.508i)2-s + (0.937 − 0.348i)3-s + (0.482 + 0.875i)4-s + (−0.998 + 0.0592i)5-s + (−0.984 − 0.176i)6-s + (0.582 + 0.812i)7-s + (0.0296 − 0.999i)8-s + (0.757 − 0.652i)9-s + (0.889 + 0.456i)10-s + (0.674 − 0.737i)11-s + (0.757 + 0.652i)12-s + (0.375 + 0.926i)13-s + (−0.0887 − 0.996i)14-s + (−0.915 + 0.403i)15-s + (−0.533 + 0.845i)16-s + (0.972 − 0.234i)17-s + ⋯ |
L(s) = 1 | + (−0.861 − 0.508i)2-s + (0.937 − 0.348i)3-s + (0.482 + 0.875i)4-s + (−0.998 + 0.0592i)5-s + (−0.984 − 0.176i)6-s + (0.582 + 0.812i)7-s + (0.0296 − 0.999i)8-s + (0.757 − 0.652i)9-s + (0.889 + 0.456i)10-s + (0.674 − 0.737i)11-s + (0.757 + 0.652i)12-s + (0.375 + 0.926i)13-s + (−0.0887 − 0.996i)14-s + (−0.915 + 0.403i)15-s + (−0.533 + 0.845i)16-s + (0.972 − 0.234i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.835 - 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.835 - 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8633419657 - 0.2582726637i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8633419657 - 0.2582726637i\) |
\(L(1)\) |
\(\approx\) |
\(0.8911449009 - 0.2091979000i\) |
\(L(1)\) |
\(\approx\) |
\(0.8911449009 - 0.2091979000i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 107 | \( 1 \) |
good | 2 | \( 1 + (-0.861 - 0.508i)T \) |
| 3 | \( 1 + (0.937 - 0.348i)T \) |
| 5 | \( 1 + (-0.998 + 0.0592i)T \) |
| 7 | \( 1 + (0.582 + 0.812i)T \) |
| 11 | \( 1 + (0.674 - 0.737i)T \) |
| 13 | \( 1 + (0.375 + 0.926i)T \) |
| 17 | \( 1 + (0.972 - 0.234i)T \) |
| 19 | \( 1 + (-0.717 - 0.696i)T \) |
| 23 | \( 1 + (-0.0887 + 0.996i)T \) |
| 29 | \( 1 + (-0.205 - 0.978i)T \) |
| 31 | \( 1 + (0.263 - 0.964i)T \) |
| 37 | \( 1 + (0.147 + 0.989i)T \) |
| 41 | \( 1 + (-0.794 + 0.606i)T \) |
| 43 | \( 1 + (-0.998 - 0.0592i)T \) |
| 47 | \( 1 + (-0.794 - 0.606i)T \) |
| 53 | \( 1 + (-0.861 + 0.508i)T \) |
| 59 | \( 1 + (-0.205 + 0.978i)T \) |
| 61 | \( 1 + (0.582 - 0.812i)T \) |
| 67 | \( 1 + (0.0296 + 0.999i)T \) |
| 71 | \( 1 + (0.937 + 0.348i)T \) |
| 73 | \( 1 + (-0.956 - 0.292i)T \) |
| 79 | \( 1 + (-0.430 + 0.902i)T \) |
| 83 | \( 1 + (-0.320 - 0.947i)T \) |
| 89 | \( 1 + (-0.984 + 0.176i)T \) |
| 97 | \( 1 + (0.889 + 0.456i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−29.97881765003021532830120595527, −28.061284758539666789932906197702, −27.43418978125446807537487207345, −26.82170987698121377670556892785, −25.681844929638692692681489757901, −24.895279533131967804376264772310, −23.73714271211976824205913885901, −22.84187898176313826554047014760, −20.83150536709116489940720285969, −20.16197235333156040425761150773, −19.40782022585684661386113745104, −18.281307831600135194784857034053, −16.92679115578867154822424456959, −16.00273725971296493048142846336, −14.8129921389250142674006175290, −14.393446123516684078107154843207, −12.50698433983187814436781663763, −10.860795634419125071934150212387, −10.07105666353398455758654191723, −8.56148176002512797237844235766, −7.92739971598855134099998521986, −6.92720366766697960688047223848, −4.82590221923622177983864427221, −3.52860533953620700231919518441, −1.494010531897157102693682615336,
1.48653297622429575440924865355, 2.95794098381527022066430297614, 4.08904953439529524681511515370, 6.62713918422086665468229760163, 7.93402000249816549546707062104, 8.57820540305017769549138664530, 9.586219429521271832084240592066, 11.458823734911026795779843012434, 11.86220167313765460061371900481, 13.36326953390912457815757516955, 14.80296876675989326721555730615, 15.74428368089458659657579275610, 17.018574281971032708756631148728, 18.62720370164514090971603059617, 18.93809886042448155833758547748, 19.88887859429835470570855321044, 21.0167385943986320299297956125, 21.802368964181929492360292376368, 23.67944289901507339887578344164, 24.615989795583764448840881802557, 25.60284500056244765546265649312, 26.60109928437692560856488196762, 27.466850665701619367740085270861, 28.221221849259555183571614305282, 29.75244209099276382557553994925