L(s) = 1 | − 8·4-s + 52·13-s − 69·16-s − 62·19-s − 398·25-s − 82·31-s − 1.13e3·37-s − 1.56e3·43-s − 416·52-s + 886·61-s + 766·64-s − 2.08e3·67-s − 2.39e3·73-s + 496·76-s + 984·79-s − 682·97-s + 3.18e3·100-s − 3.02e3·103-s − 2.47e3·109-s − 1.94e3·121-s + 656·124-s + 127-s + 131-s + 137-s + 139-s + 9.05e3·148-s + 149-s + ⋯ |
L(s) = 1 | − 4-s + 1.10·13-s − 1.07·16-s − 0.748·19-s − 3.18·25-s − 0.475·31-s − 5.02·37-s − 5.55·43-s − 1.10·52-s + 1.85·61-s + 1.49·64-s − 3.80·67-s − 3.84·73-s + 0.748·76-s + 1.40·79-s − 0.713·97-s + 3.18·100-s − 2.89·103-s − 2.17·109-s − 1.46·121-s + 0.475·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 5.02·148-s + 0.000549·149-s + ⋯ |
Λ(s)=(=((318⋅712)s/2ΓC(s)6L(s)Λ(4−s)
Λ(s)=(=((318⋅712)s/2ΓC(s+3/2)6L(s)Λ(1−s)
Particular Values
L(2) |
= |
0 |
L(21) |
= |
0 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 7 | 1 |
good | 2 | 1+p3T2+133T4+425pT6+133p6T8+p15T10+p18T12 |
| 5 | 1+398T2+88267T4+13515052T6+88267p6T8+398p12T10+p18T12 |
| 11 | 1+1946T2+1291819T4−782053340T6+1291819p6T8+1946p12T10+p18T12 |
| 13 | (1−2pT+20p2T2−150062T3+20p5T4−2p7T5+p9T6)2 |
| 17 | 1+19638T2+193457043T4+1185167951260T6+193457043p6T8+19638p12T10+p18T12 |
| 19 | (1+31T+11177T2+354214T3+11177p3T4+31p6T5+p9T6)2 |
| 23 | 1+26874T2+137157615T4−1070796212084T6+137157615p6T8+26874p12T10+p18T12 |
| 29 | 1+107486T2+5558833147T4+171068655050380T6+5558833147p6T8+107486p12T10+p18T12 |
| 31 | (1+41T+55666T2+237883T3+55666p3T4+41p6T5+p9T6)2 |
| 37 | (1+566T+246688T2+61584934T3+246688p3T4+566p6T5+p9T6)2 |
| 41 | 1+201446T2+24090872899T4+1992163622083516T6+24090872899p6T8+201446p12T10+p18T12 |
| 43 | (1+783T+433368T2+140027659T3+433368p3T4+783p6T5+p9T6)2 |
| 47 | 1+127818T2+18065263059T4+2679796621113892T6+18065263059p6T8+127818p12T10+p18T12 |
| 53 | 1+680526T2+219567635127T4+41561598578306884T6+219567635127p6T8+680526p12T10+p18T12 |
| 59 | 1+252810T2+118459786923T4+17779226383468420T6+118459786923p6T8+252810p12T10+p18T12 |
| 61 | (1−443T+203324T2+18833347T3+203324p3T4−443p6T5+p9T6)2 |
| 67 | (1+1042T+867758T2+445253716T3+867758p3T4+1042p6T5+p9T6)2 |
| 71 | 1+1054578T2+491902090755T4+2420618974688108pT6+491902090755p6T8+1054578p12T10+p18T12 |
| 73 | (1+1199T+1316443T2+794269678T3+1316443p3T4+1199p6T5+p9T6)2 |
| 79 | (1−492T+608946T2−546599468T3+608946p3T4−492p6T5+p9T6)2 |
| 83 | 1−834486T2+1207789948167T4−565854790583996660T6+1207789948167p6T8−834486p12T10+p18T12 |
| 89 | 1+949814T2+691701373123T4+216133792389242716T6+691701373123p6T8+949814p12T10+p18T12 |
| 97 | (1+341T+490090T2−960147395T3+490090p3T4+341p6T5+p9T6)2 |
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L(s)=p∏ j=1∏12(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−5.26294724029638900132206161947, −4.85451331151224034712813048845, −4.73078665708286700704571438026, −4.65583043705429639970561919243, −4.49871053815137030683181201711, −4.46231177138341515217118411778, −4.25078958207615032378551319388, −3.82758166150088820172536582103, −3.78121247416154817245925447508, −3.76514500587472660682676713347, −3.59123432968280780572248451908, −3.38065834222808802547182986241, −3.36458530371851516773485364460, −3.13131109833473487007797049253, −2.86373177777671029302926768517, −2.59047028250162005532818507251, −2.48602901327919660557542862417, −2.03878263989753877803479575372, −2.01668180266059204247380194139, −1.95461313810555395240566264092, −1.60981621940311158152830876499, −1.52633736250609528190850799819, −1.33146412243848291458767386932, −1.19670716109598163394732490012, −0.900100721732320072677139453255, 0, 0, 0, 0, 0, 0,
0.900100721732320072677139453255, 1.19670716109598163394732490012, 1.33146412243848291458767386932, 1.52633736250609528190850799819, 1.60981621940311158152830876499, 1.95461313810555395240566264092, 2.01668180266059204247380194139, 2.03878263989753877803479575372, 2.48602901327919660557542862417, 2.59047028250162005532818507251, 2.86373177777671029302926768517, 3.13131109833473487007797049253, 3.36458530371851516773485364460, 3.38065834222808802547182986241, 3.59123432968280780572248451908, 3.76514500587472660682676713347, 3.78121247416154817245925447508, 3.82758166150088820172536582103, 4.25078958207615032378551319388, 4.46231177138341515217118411778, 4.49871053815137030683181201711, 4.65583043705429639970561919243, 4.73078665708286700704571438026, 4.85451331151224034712813048845, 5.26294724029638900132206161947
Plot not available for L-functions of degree greater than 10.