L(s) = 1 | + 2·2-s − 2·3-s + 3·4-s + 8·5-s − 4·6-s − 2·7-s + 4·8-s + 5·9-s + 16·10-s − 2·11-s − 6·12-s − 2·13-s − 4·14-s − 16·15-s + 4·16-s − 6·17-s + 10·18-s − 6·19-s + 24·20-s + 4·21-s − 4·22-s − 8·23-s − 8·24-s − 4·25-s − 4·26-s − 6·27-s − 6·28-s + ⋯ |
L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s + 3.57·5-s − 1.63·6-s − 0.755·7-s + 1.41·8-s + 5/3·9-s + 5.05·10-s − 0.603·11-s − 1.73·12-s − 0.554·13-s − 1.06·14-s − 4.13·15-s + 16-s − 1.45·17-s + 2.35·18-s − 1.37·19-s + 5.36·20-s + 0.872·21-s − 0.852·22-s − 1.66·23-s − 1.63·24-s − 4/5·25-s − 0.784·26-s − 1.15·27-s − 1.13·28-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) |
≈ |
0.9163777444 |
L(21) |
≈ |
0.9163777444 |
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+T2+T4−p4T5+25T6−3pT7−3T8−3p2T9+25p2T10−p7T11+p4T12+p6T14−p8T15+p8T16 |
| 3 | 1+2T−T2−2pT3−4T4+4p2T5+65T6−20T7−113T8−20pT9+65p2T10+4p5T11−4p4T12−2p6T13−p6T14+2p7T15+p8T16 |
| 5 | (1−T+pT2)8 |
| 7 | 1+2T−9T2−30T3+36T4+156T5−135T6−324T7+1527T8−324pT9−135p2T10+156p3T11+36p4T12−30p5T13−9p6T14+2p7T15+p8T16 |
| 11 | 1+2T−T2+10T3−52T4+820T5+2241T6−1972T7+9183T8−1972pT9+2241p2T10+820p3T11−52p4T12+10p5T13−p6T14+2p7T15+p8T16 |
| 13 | 1+2T−15T2−42T3+84T4+996T5+1839T6−5964T7−35169T8−5964pT9+1839p2T10+996p3T11+84p4T12−42p5T13−15p6T14+2p7T15+p8T16 |
| 17 | 1+6T+T2−6pT3−324T4+84pT5+9215T6+1740T7−72793T8+1740pT9+9215p2T10+84p4T11−324p4T12−6p6T13+p6T14+6p7T15+p8T16 |
| 19 | 1+6T−9T2−210T3−28pT4−900T5−2151T6+42756T7+442783T8+42756pT9−2151p2T10−900p3T11−28p5T12−210p5T13−9p6T14+6p7T15+p8T16 |
| 23 | (1+4T−7T2−120T3−319T4−120pT5−7p2T6+4p3T7+p4T8)2 |
| 29 | 1+8T−2T2−312T3−1349T4+520T5+22492T6+95040T7+530997T8+95040pT9+22492p2T10+520p3T11−1349p4T12−312p5T13−2p6T14+8p7T15+p8T16 |
| 37 | (1+T+pT2)8 |
| 41 | 1+2T−7T2+46T3−1348T4+11740T5+53847T6−209308T7+2217735T8−209308pT9+53847p2T10+11740p3T11−1348p4T12+46p5T13−7p6T14+2p7T15+p8T16 |
| 43 | 1+2T+15T2+138T3−1236T4+6996T5−15471T6−315444T7+2085951T8−315444pT9−15471p2T10+6996p3T11−1236p4T12+138p5T13+15p6T14+2p7T15+p8T16 |
| 47 | 1+8T−14T2−360T3−989T4+35464T5+256420T6−291456T7−6813243T8−291456pT9+256420p2T10+35464p3T11−989p4T12−360p5T13−14p6T14+8p7T15+p8T16 |
| 53 | 1−6T−71T2+750T3+2196T4−15468T5−181225T6−62652T7+18517247T8−62652pT9−181225p2T10−15468p3T11+2196p4T12+750p5T13−71p6T14−6p7T15+p8T16 |
| 59 | 1−6T−41T2+6pT3−324T4−1092pT5+428921T6+1269852T7−16710673T8+1269852pT9+428921p2T10−1092p4T11−324p4T12+6p6T13−41p6T14−6p7T15+p8T16 |
| 61 | (1+114T2+p2T4)4 |
| 67 | (1+2T+117T2+2pT3+p2T4)4 |
| 71 | 1−14T+55T2+210T3−1820T4−72772T5+789433T6−40740T7−29229705T8−40740pT9+789433p2T10−72772p3T11−1820p4T12+210p5T13+55p6T14−14p7T15+p8T16 |
| 73 | 1−2T−135T2+402T3+12924T4−15516T5−1104681T6+336924T7+86270151T8+336924pT9−1104681p2T10−15516p3T11+12924p4T12+402p5T13−135p6T14−2p7T15+p8T16 |
| 79 | 1+22T+223T2+902T3−6364T4−127820T5−975103T6−4781260T7−33119273T8−4781260pT9−975103p2T10−127820p3T11−6364p4T12+902p5T13+223p6T14+22p7T15+p8T16 |
| 83 | 1+6T−89T2−786T3+2508T4+114396T5+594569T6−4068348T7−59039617T8−4068348pT9+594569p2T10+114396p3T11+2508p4T12−786p5T13−89p6T14+6p7T15+p8T16 |
| 89 | 1+8T−58T2−728T3−973T4+183880T5+1530828T6−5113024T7−89035515T8−5113024pT9+1530828p2T10+183880p3T11−973p4T12−728p5T13−58p6T14+8p7T15+p8T16 |
| 97 | 1+16T+6T2−2352T3−23709T4−154992T5−259380T6+19931520T7+366279717T8+19931520pT9−259380p2T10−154992p3T11−23709p4T12−2352p5T13+6p6T14+16p7T15+p8T16 |
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L(s)=p∏ j=1∏16(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−4.35796226171341044327663549902, −4.14788155802545182707558147771, −4.02364988197992490867794547603, −3.96343562414858544794805002969, −3.86417833872985660477945235632, −3.83929580130220073589804698168, −3.78738520797133276492607401181, −3.32752993579582653685878773543, −3.26912413271597384975892766400, −2.93664624550242929592999052010, −2.91045123429347156003971823042, −2.77818011815767627582475851594, −2.38779384657354344961201066707, −2.33920794356340236606440686716, −2.27217038297573760227989382913, −2.27064705210076804052635103640, −1.97528934845772330902290957294, −1.95791126517290227916977239341, −1.76973640119156840354323532387, −1.58562171812125841949779020464, −1.57252130922744678677997649984, −1.26904479384264766825004381611, −0.926999746360359762691088287768, −0.28218391925381551714308506094, −0.12872691484090804017722086150,
0.12872691484090804017722086150, 0.28218391925381551714308506094, 0.926999746360359762691088287768, 1.26904479384264766825004381611, 1.57252130922744678677997649984, 1.58562171812125841949779020464, 1.76973640119156840354323532387, 1.95791126517290227916977239341, 1.97528934845772330902290957294, 2.27064705210076804052635103640, 2.27217038297573760227989382913, 2.33920794356340236606440686716, 2.38779384657354344961201066707, 2.77818011815767627582475851594, 2.91045123429347156003971823042, 2.93664624550242929592999052010, 3.26912413271597384975892766400, 3.32752993579582653685878773543, 3.78738520797133276492607401181, 3.83929580130220073589804698168, 3.86417833872985660477945235632, 3.96343562414858544794805002969, 4.02364988197992490867794547603, 4.14788155802545182707558147771, 4.35796226171341044327663549902
Plot not available for L-functions of degree greater than 10.