| L(s) = 1 | − 6·3-s + 14·4-s + 4·7-s − 89·9-s + 90·11-s − 84·12-s + 95·16-s − 24·21-s + 569·25-s + 564·27-s + 56·28-s − 540·33-s − 1.24e3·36-s − 372·37-s + 690·41-s + 1.26e3·44-s − 1.39e3·47-s − 570·48-s − 1.55e3·49-s + 768·53-s − 356·63-s + 276·64-s + 1.10e3·67-s − 12·71-s − 1.44e3·73-s − 3.41e3·75-s + 360·77-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 7/4·4-s + 0.215·7-s − 3.29·9-s + 2.46·11-s − 2.02·12-s + 1.48·16-s − 0.249·21-s + 4.55·25-s + 4.02·27-s + 0.377·28-s − 2.84·33-s − 5.76·36-s − 1.65·37-s + 2.62·41-s + 4.31·44-s − 4.32·47-s − 1.71·48-s − 4.54·49-s + 1.99·53-s − 0.711·63-s + 0.539·64-s + 2.00·67-s − 0.0200·71-s − 2.31·73-s − 5.25·75-s + 0.532·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(37^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(37^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.550046718\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.550046718\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + 372 T + 1496 p T^{2} + 15756 p^{2} T^{3} + 163254 p^{3} T^{4} + 15756 p^{5} T^{5} + 1496 p^{7} T^{6} + 372 p^{9} T^{7} + p^{12} T^{8} \) |
| good | 2 | \( 1 - 7 p T^{2} + 101 T^{4} - 45 p^{3} T^{6} + 185 p^{2} T^{8} - 45 p^{9} T^{10} + 101 p^{12} T^{12} - 7 p^{19} T^{14} + p^{24} T^{16} \) |
| 3 | \( ( 1 + p T + 58 T^{2} + 62 p T^{3} + 2101 T^{4} + 62 p^{4} T^{5} + 58 p^{6} T^{6} + p^{10} T^{7} + p^{12} T^{8} )^{2} \) |
| 5 | \( 1 - 569 T^{2} + 143771 T^{4} - 22531803 T^{6} + 2876521184 T^{8} - 22531803 p^{6} T^{10} + 143771 p^{12} T^{12} - 569 p^{18} T^{14} + p^{24} T^{16} \) |
| 7 | \( ( 1 - 2 T + 785 T^{2} + 402 T^{3} + 357104 T^{4} + 402 p^{3} T^{5} + 785 p^{6} T^{6} - 2 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 11 | \( ( 1 - 45 T + 4568 T^{2} - 174690 T^{3} + 8636091 T^{4} - 174690 p^{3} T^{5} + 4568 p^{6} T^{6} - 45 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 13 | \( 1 - 6665 T^{2} + 30449611 T^{4} - 95990572955 T^{6} + 242497383975136 T^{8} - 95990572955 p^{6} T^{10} + 30449611 p^{12} T^{12} - 6665 p^{18} T^{14} + p^{24} T^{16} \) |
| 17 | \( 1 - 26804 T^{2} + 355856612 T^{4} - 3010719605388 T^{6} + 17603694246450230 T^{8} - 3010719605388 p^{6} T^{10} + 355856612 p^{12} T^{12} - 26804 p^{18} T^{14} + p^{24} T^{16} \) |
| 19 | \( 1 - 38948 T^{2} + 729897364 T^{4} - 8659597849148 T^{6} + 70828219227174454 T^{8} - 8659597849148 p^{6} T^{10} + 729897364 p^{12} T^{12} - 38948 p^{18} T^{14} + p^{24} T^{16} \) |
| 23 | \( 1 - 51185 T^{2} + 1439986991 T^{4} - 26874613654539 T^{6} + 375869554389627560 T^{8} - 26874613654539 p^{6} T^{10} + 1439986991 p^{12} T^{12} - 51185 p^{18} T^{14} + p^{24} T^{16} \) |
| 29 | \( 1 - 96305 T^{2} + 5253907043 T^{4} - 190709730777051 T^{6} + 5309460523584505136 T^{8} - 190709730777051 p^{6} T^{10} + 5253907043 p^{12} T^{12} - 96305 p^{18} T^{14} + p^{24} T^{16} \) |
| 31 | \( 1 + 40387 T^{2} + 1293231943 T^{4} - 11459961908975 T^{6} - 463205768029267688 T^{8} - 11459961908975 p^{6} T^{10} + 1293231943 p^{12} T^{12} + 40387 p^{18} T^{14} + p^{24} T^{16} \) |
| 41 | \( ( 1 - 345 T + 226310 T^{2} - 60382440 T^{3} + 21595147845 T^{4} - 60382440 p^{3} T^{5} + 226310 p^{6} T^{6} - 345 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 43 | \( 1 - 298616 T^{2} + 51759036316 T^{4} - 6152954729170952 T^{6} + \)\(55\!\cdots\!30\)\( T^{8} - 6152954729170952 p^{6} T^{10} + 51759036316 p^{12} T^{12} - 298616 p^{18} T^{14} + p^{24} T^{16} \) |
| 47 | \( ( 1 + 696 T + 439637 T^{2} + 187575822 T^{3} + 67185690216 T^{4} + 187575822 p^{3} T^{5} + 439637 p^{6} T^{6} + 696 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 53 | \( ( 1 - 384 T + 405365 T^{2} - 89934390 T^{3} + 69302222916 T^{4} - 89934390 p^{3} T^{5} + 405365 p^{6} T^{6} - 384 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 59 | \( 1 - 792188 T^{2} + 367363638548 T^{4} - 118482096698958180 T^{6} + \)\(27\!\cdots\!82\)\( T^{8} - 118482096698958180 p^{6} T^{10} + 367363638548 p^{12} T^{12} - 792188 p^{18} T^{14} + p^{24} T^{16} \) |
| 61 | \( 1 - 1192409 T^{2} + 650923527259 T^{4} - 223348095401889395 T^{6} + \)\(56\!\cdots\!44\)\( T^{8} - 223348095401889395 p^{6} T^{10} + 650923527259 p^{12} T^{12} - 1192409 p^{18} T^{14} + p^{24} T^{16} \) |
| 67 | \( ( 1 - 551 T + 1060013 T^{2} - 370570431 T^{3} + 438535700444 T^{4} - 370570431 p^{3} T^{5} + 1060013 p^{6} T^{6} - 551 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 71 | \( ( 1 + 6 T + 342401 T^{2} - 355994586 T^{3} + 35784043512 T^{4} - 355994586 p^{3} T^{5} + 342401 p^{6} T^{6} + 6 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 73 | \( ( 1 + 721 T + 1255238 T^{2} + 719772432 T^{3} + 709684675457 T^{4} + 719772432 p^{3} T^{5} + 1255238 p^{6} T^{6} + 721 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 79 | \( 1 - 1088321 T^{2} + 944344757791 T^{4} - 454030831289708867 T^{6} + \)\(25\!\cdots\!36\)\( T^{8} - 454030831289708867 p^{6} T^{10} + 944344757791 p^{12} T^{12} - 1088321 p^{18} T^{14} + p^{24} T^{16} \) |
| 83 | \( ( 1 - 846 T + 1978133 T^{2} - 1245383370 T^{3} + 1665115135116 T^{4} - 1245383370 p^{3} T^{5} + 1978133 p^{6} T^{6} - 846 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 89 | \( 1 - 3661640 T^{2} + 6912886298300 T^{4} - 8352114070134388728 T^{6} + \)\(70\!\cdots\!86\)\( T^{8} - 8352114070134388728 p^{6} T^{10} + 6912886298300 p^{12} T^{12} - 3661640 p^{18} T^{14} + p^{24} T^{16} \) |
| 97 | \( 1 - 2630300 T^{2} + 2969290197028 T^{4} - 2108146670518191140 T^{6} + \)\(15\!\cdots\!30\)\( T^{8} - 2108146670518191140 p^{6} T^{10} + 2969290197028 p^{12} T^{12} - 2630300 p^{18} T^{14} + p^{24} T^{16} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.25307284704792828827910614884, −7.08788874556800578417784776865, −6.93528086878899315201948435406, −6.77530202294622556953426388670, −6.71529562665765827338132586161, −6.52997025386277423716920602153, −6.33368593320922421835413289436, −6.04027819712113273664708871590, −6.02807938425407256996016691887, −5.63352014928260157533404528442, −5.46073927816365930619851125771, −5.33200618577179860286216396053, −4.98234314011458615500957577835, −4.77915149559462255445209375472, −4.62777502386002529169897026901, −4.22204746749457819931203692813, −3.62327125316605440196122914665, −3.39205103370227674598536145932, −3.25631944161266833888874095101, −2.84209136377387485282669692008, −2.77941169501002676816854013125, −2.34157561726656982925440354486, −1.46505892223313516241736175532, −1.42947823421608526138806451791, −0.42688964957054755265736946320,
0.42688964957054755265736946320, 1.42947823421608526138806451791, 1.46505892223313516241736175532, 2.34157561726656982925440354486, 2.77941169501002676816854013125, 2.84209136377387485282669692008, 3.25631944161266833888874095101, 3.39205103370227674598536145932, 3.62327125316605440196122914665, 4.22204746749457819931203692813, 4.62777502386002529169897026901, 4.77915149559462255445209375472, 4.98234314011458615500957577835, 5.33200618577179860286216396053, 5.46073927816365930619851125771, 5.63352014928260157533404528442, 6.02807938425407256996016691887, 6.04027819712113273664708871590, 6.33368593320922421835413289436, 6.52997025386277423716920602153, 6.71529562665765827338132586161, 6.77530202294622556953426388670, 6.93528086878899315201948435406, 7.08788874556800578417784776865, 7.25307284704792828827910614884
Plot not available for L-functions of degree greater than 10.
