L(s) = 1 | + (−0.648 − 0.761i)2-s + (0.613 + 0.789i)3-s + (−0.158 + 0.987i)4-s + (0.538 − 0.842i)5-s + (0.203 − 0.979i)6-s + (0.746 − 0.665i)7-s + (0.854 − 0.519i)8-s + (−0.247 + 0.968i)9-s + (−0.990 + 0.136i)10-s + (−0.974 + 0.225i)11-s + (−0.877 + 0.480i)12-s + (0.377 − 0.926i)13-s + (−0.990 − 0.136i)14-s + (0.995 − 0.0909i)15-s + (−0.949 − 0.313i)16-s + (0.113 + 0.993i)17-s + ⋯ |
L(s) = 1 | + (−0.648 − 0.761i)2-s + (0.613 + 0.789i)3-s + (−0.158 + 0.987i)4-s + (0.538 − 0.842i)5-s + (0.203 − 0.979i)6-s + (0.746 − 0.665i)7-s + (0.854 − 0.519i)8-s + (−0.247 + 0.968i)9-s + (−0.990 + 0.136i)10-s + (−0.974 + 0.225i)11-s + (−0.877 + 0.480i)12-s + (0.377 − 0.926i)13-s + (−0.990 − 0.136i)14-s + (0.995 − 0.0909i)15-s + (−0.949 − 0.313i)16-s + (0.113 + 0.993i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.817 - 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.817 - 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.019869602 - 0.3236023975i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.019869602 - 0.3236023975i\) |
\(L(1)\) |
\(\approx\) |
\(1.001658906 - 0.2257161215i\) |
\(L(1)\) |
\(\approx\) |
\(1.001658906 - 0.2257161215i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 139 | \( 1 \) |
good | 2 | \( 1 + (-0.648 - 0.761i)T \) |
| 3 | \( 1 + (0.613 + 0.789i)T \) |
| 5 | \( 1 + (0.538 - 0.842i)T \) |
| 7 | \( 1 + (0.746 - 0.665i)T \) |
| 11 | \( 1 + (-0.974 + 0.225i)T \) |
| 13 | \( 1 + (0.377 - 0.926i)T \) |
| 17 | \( 1 + (0.113 + 0.993i)T \) |
| 19 | \( 1 + (0.803 - 0.595i)T \) |
| 23 | \( 1 + (0.203 + 0.979i)T \) |
| 29 | \( 1 + (0.934 - 0.356i)T \) |
| 31 | \( 1 + (-0.247 - 0.968i)T \) |
| 37 | \( 1 + (0.995 + 0.0909i)T \) |
| 41 | \( 1 + (-0.829 + 0.557i)T \) |
| 43 | \( 1 + (-0.5 + 0.866i)T \) |
| 47 | \( 1 + (-0.998 + 0.0455i)T \) |
| 53 | \( 1 + (0.983 - 0.181i)T \) |
| 59 | \( 1 + (-0.775 + 0.631i)T \) |
| 61 | \( 1 + (-0.829 - 0.557i)T \) |
| 67 | \( 1 + (0.291 + 0.956i)T \) |
| 71 | \( 1 + (-0.998 - 0.0455i)T \) |
| 73 | \( 1 + (0.0227 + 0.999i)T \) |
| 79 | \( 1 + (-0.0682 + 0.997i)T \) |
| 83 | \( 1 + (0.291 - 0.956i)T \) |
| 89 | \( 1 + (-0.419 + 0.907i)T \) |
| 97 | \( 1 + (-0.5 - 0.866i)T \) |
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
−28.77120437754632197270261647683, −27.07785023493026008307503023959, −26.45673237875827083801670932237, −25.450604762059465885721955211584, −24.843440819042415919790073786786, −23.864667355984321926577157456, −22.95011723941905042457595672555, −21.42192499266609738058136162717, −20.36503717865789547860549822538, −18.8838154715277671913194587015, −18.36471628967476117715009863001, −17.902385268936554323716216461860, −16.33186083274742893223579758385, −15.125770350066310073947484629042, −14.21803990972927001837974613488, −13.64796599925520770862110042746, −11.877563788551183368680357086155, −10.609878444005561898777757265288, −9.314713714462719021560705670521, −8.35208701978195629393107797102, −7.32601216777571603082933605472, −6.365175922532507579939037802, −5.17955676299367176246874895155, −2.78970241800130730123324400297, −1.63578336842254283067396660257,
1.4022832704378011803198696479, 2.84826216081643360075193154551, 4.23646486539278225330008757690, 5.27473848789193854228361109492, 7.80951482286307131768031022974, 8.33474981087587966451204472361, 9.65122406290527003705880791504, 10.34078836642340838472321304667, 11.385467529705183591696320677271, 13.06676110450190398719369856800, 13.575058085151831504339540551219, 15.20365659951080279025787127061, 16.33609120667692777980108530789, 17.31673312189779598192642980112, 18.12865862493154097369854977115, 19.77692456514969001106529132795, 20.30564358831127812792005196915, 21.13599215748935617913205939567, 21.72453650045880590196860109644, 23.267386996102670497920917268652, 24.6930965149160201161462047753, 25.72536183837466846695910847927, 26.43765144494909353163859912290, 27.52204577149853516880320372564, 28.126150864524528786232960206311