L(s) = 1 | + 2·2-s − 3·3-s − 2·4-s + 3·5-s − 6·6-s − 2·7-s − 11·8-s − 5·9-s + 6·10-s − 18·11-s + 6·12-s − 8·13-s − 4·14-s − 9·15-s − 11·16-s − 14·17-s − 10·18-s − 6·19-s − 6·20-s + 6·21-s − 36·22-s − 22·23-s + 33·24-s − 9·25-s − 16·26-s + 21·27-s + 4·28-s + ⋯ |
L(s) = 1 | + 1.41·2-s − 1.73·3-s − 4-s + 1.34·5-s − 2.44·6-s − 0.755·7-s − 3.88·8-s − 5/3·9-s + 1.89·10-s − 5.42·11-s + 1.73·12-s − 2.21·13-s − 1.06·14-s − 2.32·15-s − 2.75·16-s − 3.39·17-s − 2.35·18-s − 1.37·19-s − 1.34·20-s + 1.30·21-s − 7.67·22-s − 4.58·23-s + 6.73·24-s − 9/5·25-s − 3.13·26-s + 4.04·27-s + 0.755·28-s + ⋯ |
Λ(s)=(=((3116)s/2ΓC(s)8L(s)Λ(2−s)
Λ(s)=(=((3116)s/2ΓC(s+1/2)8L(s)Λ(1−s)
Particular Values
L(1) |
= |
0 |
L(21) |
= |
0 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 31 | 1 |
good | 2 | 1−pT+3pT2−5T3+11T4−T5+15T6+9T7+27T8+9pT9+15p2T10−p3T11+11p4T12−5p5T13+3p7T14−p8T15+p8T16 |
| 3 | 1+pT+14T2+4p2T3+37pT4+26p2T5+560T6+335pT7+1987T8+335p2T9+560p2T10+26p5T11+37p5T12+4p7T13+14p6T14+p8T15+p8T16 |
| 5 | 1−3T+18T2−36T3+31pT4−261T5+993T6−1428T7+5151T8−1428pT9+993p2T10−261p3T11+31p5T12−36p5T13+18p6T14−3p7T15+p8T16 |
| 7 | 1+2T+31T2+60T3+506T4+881T5+5550T6+8441T7+44837T8+8441pT9+5550p2T10+881p3T11+506p4T12+60p5T13+31p6T14+2p7T15+p8T16 |
| 11 | 1+18T+219T2+1875T3+12998T4+73485T5+354666T6+132522pT7+5210553T8+132522p2T9+354666p2T10+73485p3T11+12998p4T12+1875p5T13+219p6T14+18p7T15+p8T16 |
| 13 | 1+8T+100T2+567T3+4139T4+18554T5+100344T6+368819T7+1591961T8+368819pT9+100344p2T10+18554p3T11+4139p4T12+567p5T13+100p6T14+8p7T15+p8T16 |
| 17 | 1+14T+171T2+1412T3+10571T4+64147T5+358770T6+1712910T7+7592697T8+1712910pT9+358770p2T10+64147p3T11+10571p4T12+1412p5T13+171p6T14+14p7T15+p8T16 |
| 19 | 1+6T+126T2+550T3+6723T4+22465T5+213499T6+577881T7+4727383T8+577881pT9+213499p2T10+22465p3T11+6723p4T12+550p5T13+126p6T14+6p7T15+p8T16 |
| 23 | 1+22T+372T2+4321T3+42281T4+334832T5+2312310T6+13546893T7+70125249T8+13546893pT9+2312310p2T10+334832p3T11+42281p4T12+4321p5T13+372p6T14+22p7T15+p8T16 |
| 29 | 1+12T+228T2+2067T3+22391T4+160170T5+1262877T6+7289670T7+45291747T8+7289670pT9+1262877p2T10+160170p3T11+22391p4T12+2067p5T13+228p6T14+12p7T15+p8T16 |
| 37 | 1−8T+184T2−1003T3+14093T4−50978T5+642206T6−1564129T7+23810023T8−1564129pT9+642206p2T10−50978p3T11+14093p4T12−1003p5T13+184p6T14−8p7T15+p8T16 |
| 41 | 1+22T+498T2+6811T3+89027T4+877490T5+8169387T6+61298622T7+433574745T8+61298622pT9+8169387p2T10+877490p3T11+89027p4T12+6811p5T13+498p6T14+22p7T15+p8T16 |
| 43 | 1−2T+205T2−403T3+21134T4−40961T5+1453424T6−2649346T7+72598441T8−2649346pT9+1453424p2T10−40961p3T11+21134p4T12−403p5T13+205p6T14−2p7T15+p8T16 |
| 47 | 1+18T+411T2+4965T3+66911T4+618324T5+6152625T6+45556584T7+358374597T8+45556584pT9+6152625p2T10+618324p3T11+66911p4T12+4965p5T13+411p6T14+18p7T15+p8T16 |
| 53 | 1+6T+244T2+1605T3+30376T4+203598T5+2505355T6+16037007T7+152391427T8+16037007pT9+2505355p2T10+203598p3T11+30376p4T12+1605p5T13+244p6T14+6p7T15+p8T16 |
| 59 | 1+4T+309T2+1189T3+45401T4+174557T5+4316301T6+15922002T7+295944207T8+15922002pT9+4316301p2T10+174557p3T11+45401p4T12+1189p5T13+309p6T14+4p7T15+p8T16 |
| 61 | 1+30T+776T2+13665T3+208005T4+2593710T5+28390399T6+268109040T7+2240276459T8+268109040pT9+28390399p2T10+2593710p3T11+208005p4T12+13665p5T13+776p6T14+30p7T15+p8T16 |
| 67 | 1+13T+427T2+4781T3+85148T4+816694T5+10335824T6+84073193T7+837177553T8+84073193pT9+10335824p2T10+816694p3T11+85148p4T12+4781p5T13+427p6T14+13p7T15+p8T16 |
| 71 | 1+T+465T2+325T3+99635T4+48353T5+12913848T6+4606650T7+1111294875T8+4606650pT9+12913848p2T10+48353p3T11+99635p4T12+325p5T13+465p6T14+p7T15+p8T16 |
| 73 | 1+2T+280T2+423T3+39254T4+42716T5+3669099T6+2910611T7+282548711T8+2910611pT9+3669099p2T10+42716p3T11+39254p4T12+423p5T13+280p6T14+2p7T15+p8T16 |
| 79 | 1+8T+343T2+2748T3+65666T4+478310T5+8349177T6+54496370T7+769049897T8+54496370pT9+8349177p2T10+478310p3T11+65666p4T12+2748p5T13+343p6T14+8p7T15+p8T16 |
| 83 | 1+39T+12pT2+18231T3+273488T4+3449334T5+38635404T6+390596853T7+3688239693T8+390596853pT9+38635404p2T10+3449334p3T11+273488p4T12+18231p5T13+12p7T14+39p7T15+p8T16 |
| 89 | 1+27T+807T2+15033T3+266447T4+3727035T5+48700278T6+532857324T7+5459588145T8+532857324pT9+48700278p2T10+3727035p3T11+266447p4T12+15033p5T13+807p6T14+27p7T15+p8T16 |
| 97 | 1−34T+1021T2−20487T3+372881T4−5460517T5+73647420T6−843517135T7+8954673947T8−843517135pT9+73647420p2T10−5460517p3T11+372881p4T12−20487p5T13+1021p6T14−34p7T15+p8T16 |
show more | |
show less | |
L(s)=p∏ j=1∏16(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−4.76215951198312336723481108109, −4.75635777463472870802826509276, −4.67819292569890196664591076752, −4.55391159967375220447126906130, −4.49168103786802189546101103512, −4.45312723315692809986745786382, −4.23105578159503209608378092609, −3.86546062320237866549764136901, −3.75412114503109892793723639524, −3.68908864272905324600337928599, −3.44517747570623331972260076133, −3.34003095594142756711301675214, −3.26854942024922063575180897825, −3.24838054942234849819621508955, −2.84786392983352421331922844007, −2.66617749922296601698038303947, −2.66033597440844393867383850860, −2.46566911955483183659778538062, −2.39710487264412169029959229348, −2.35378704719709710375129019762, −2.15009586201185655449189305626, −1.88638246550426006134732610674, −1.77682932002744698492454962367, −1.74651535124718702871677050199, −1.53693472626597640772997237449, 0, 0, 0, 0, 0, 0, 0, 0,
1.53693472626597640772997237449, 1.74651535124718702871677050199, 1.77682932002744698492454962367, 1.88638246550426006134732610674, 2.15009586201185655449189305626, 2.35378704719709710375129019762, 2.39710487264412169029959229348, 2.46566911955483183659778538062, 2.66033597440844393867383850860, 2.66617749922296601698038303947, 2.84786392983352421331922844007, 3.24838054942234849819621508955, 3.26854942024922063575180897825, 3.34003095594142756711301675214, 3.44517747570623331972260076133, 3.68908864272905324600337928599, 3.75412114503109892793723639524, 3.86546062320237866549764136901, 4.23105578159503209608378092609, 4.45312723315692809986745786382, 4.49168103786802189546101103512, 4.55391159967375220447126906130, 4.67819292569890196664591076752, 4.75635777463472870802826509276, 4.76215951198312336723481108109
Plot not available for L-functions of degree greater than 10.