| L(s) = 1 | − 6·3-s + 3·4-s − 16·7-s + 9·9-s − 18·12-s + 8·13-s + 16·16-s − 12·19-s + 96·21-s − 6·25-s − 48·28-s + 52·31-s + 27·36-s − 60·37-s − 48·39-s − 336·43-s − 96·48-s + 162·49-s + 24·52-s + 72·57-s + 24·61-s − 144·63-s + 16·67-s − 148·73-s + 36·75-s − 36·76-s − 80·79-s + ⋯ |
| L(s) = 1 | − 2·3-s + 3/4·4-s − 2.28·7-s + 9-s − 3/2·12-s + 8/13·13-s + 16-s − 0.631·19-s + 32/7·21-s − 0.239·25-s − 1.71·28-s + 1.67·31-s + 3/4·36-s − 1.62·37-s − 1.23·39-s − 7.81·43-s − 2·48-s + 3.30·49-s + 6/13·52-s + 1.26·57-s + 0.393·61-s − 2.28·63-s + 0.238·67-s − 2.02·73-s + 0.479·75-s − 0.473·76-s − 1.01·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2205704412\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2205704412\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 + p T + p^{2} T^{2} + p^{3} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | \( 1 \) |
| good | 2 | \( ( 1 - 5 T + 11 T^{2} - 25 T^{3} + 61 T^{4} - 25 p^{2} T^{5} + 11 p^{4} T^{6} - 5 p^{6} T^{7} + p^{8} T^{8} )( 1 + 5 T + 11 T^{2} + 25 T^{3} + 61 T^{4} + 25 p^{2} T^{5} + 11 p^{4} T^{6} + 5 p^{6} T^{7} + p^{8} T^{8} ) \) |
| 5 | \( 1 + 6 T^{2} - 589 T^{4} - 7284 T^{6} + 324421 T^{8} - 7284 p^{4} T^{10} - 589 p^{8} T^{12} + 6 p^{12} T^{14} + p^{16} T^{16} \) |
| 7 | \( ( 1 + 8 T + 15 T^{2} - 272 T^{3} - 2911 T^{4} - 272 p^{2} T^{5} + 15 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 - 4 T - 153 T^{2} + 1288 T^{3} + 20705 T^{4} + 1288 p^{2} T^{5} - 153 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 - 70 T + 2651 T^{2} - 68600 T^{3} + 1327141 T^{4} - 68600 p^{2} T^{5} + 2651 p^{4} T^{6} - 70 p^{6} T^{7} + p^{8} T^{8} )( 1 + 70 T + 2651 T^{2} + 68600 T^{3} + 1327141 T^{4} + 68600 p^{2} T^{5} + 2651 p^{4} T^{6} + 70 p^{6} T^{7} + p^{8} T^{8} ) \) |
| 19 | \( ( 1 + 6 T - 325 T^{2} - 4116 T^{3} + 92629 T^{4} - 4116 p^{2} T^{5} - 325 p^{4} T^{6} + 6 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 23 | \( ( 1 - 1014 T^{2} + p^{4} T^{4} )^{4} \) |
| 29 | \( 1 + 98 T^{2} - 697677 T^{4} - 137685884 T^{6} + 479960469605 T^{8} - 137685884 p^{4} T^{10} - 697677 p^{8} T^{12} + 98 p^{12} T^{14} + p^{16} T^{16} \) |
| 31 | \( ( 1 - 26 T - 285 T^{2} + 32396 T^{3} - 568411 T^{4} + 32396 p^{2} T^{5} - 285 p^{4} T^{6} - 26 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 + 30 T - 469 T^{2} - 55140 T^{3} - 1012139 T^{4} - 55140 p^{2} T^{5} - 469 p^{4} T^{6} + 30 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 41 | \( 1 + 3186 T^{2} + 7324835 T^{4} + 14334049764 T^{6} + 24970049473669 T^{8} + 14334049764 p^{4} T^{10} + 7324835 p^{8} T^{12} + 3186 p^{12} T^{14} + p^{16} T^{16} \) |
| 43 | \( ( 1 + 42 T + p^{2} T^{2} )^{8} \) |
| 47 | \( 1 - 3018 T^{2} + 4228643 T^{4} + 1964832684 T^{6} - 26564293943195 T^{8} + 1964832684 p^{4} T^{10} + 4228643 p^{8} T^{12} - 3018 p^{12} T^{14} + p^{16} T^{16} \) |
| 53 | \( 1 + 2054 T^{2} - 3671565 T^{4} - 23748442484 T^{6} - 19808886989371 T^{8} - 23748442484 p^{4} T^{10} - 3671565 p^{8} T^{12} + 2054 p^{12} T^{14} + p^{16} T^{16} \) |
| 59 | \( 1 + 2562 T^{2} - 5553517 T^{4} - 45272789436 T^{6} - 48694916226395 T^{8} - 45272789436 p^{4} T^{10} - 5553517 p^{8} T^{12} + 2562 p^{12} T^{14} + p^{16} T^{16} \) |
| 61 | \( ( 1 - 12 T - 3577 T^{2} + 87576 T^{3} + 12259105 T^{4} + 87576 p^{2} T^{5} - 3577 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 - 2 T + p^{2} T^{2} )^{8} \) |
| 71 | \( 1 + 6518 T^{2} + 17072643 T^{4} - 54353849684 T^{6} - 788122949983195 T^{8} - 54353849684 p^{4} T^{10} + 17072643 p^{8} T^{12} + 6518 p^{12} T^{14} + p^{16} T^{16} \) |
| 73 | \( ( 1 + 74 T + 147 T^{2} - 383468 T^{3} - 29159995 T^{4} - 383468 p^{2} T^{5} + 147 p^{4} T^{6} + 74 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 + 40 T - 4641 T^{2} - 435280 T^{3} + 11553281 T^{4} - 435280 p^{2} T^{5} - 4641 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 83 | \( 1 + 12194 T^{2} + 101235315 T^{4} + 655756664836 T^{6} + 3191838695204069 T^{8} + 655756664836 p^{4} T^{10} + 101235315 p^{8} T^{12} + 12194 p^{12} T^{14} + p^{16} T^{16} \) |
| 89 | \( ( 1 - 1586 T^{2} + p^{4} T^{4} )^{4} \) |
| 97 | \( ( 1 + 62 T - 5565 T^{2} - 928388 T^{3} - 5198971 T^{4} - 928388 p^{2} T^{5} - 5565 p^{4} T^{6} + 62 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
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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
−4.69505653065804084253686907256, −4.68845270593393339994967509620, −4.59039632014148456193568343942, −4.54047235218903439872826938531, −4.48105422986129119929578883854, −3.99304683954337036343731870272, −3.91926548229564893177850368276, −3.59712447432829192917643363262, −3.47584398137650241764381898772, −3.36422530565556940997342845762, −3.28757368619713100554309439841, −3.18160910696266606170493430349, −3.15027772003636097619838016139, −2.92942368087034259985950863045, −2.63127881566113622619271851729, −2.25905054736044839771886315695, −2.09523237694257881029268506584, −2.00815729735403313241855426868, −1.66892633947050941979797349487, −1.61164901979040926132957937197, −1.36965477048823172204803280569, −0.811292809446802322404575139449, −0.68714535341500400824359264860, −0.33062370104172192175920274758, −0.12978555830889819402369735301,
0.12978555830889819402369735301, 0.33062370104172192175920274758, 0.68714535341500400824359264860, 0.811292809446802322404575139449, 1.36965477048823172204803280569, 1.61164901979040926132957937197, 1.66892633947050941979797349487, 2.00815729735403313241855426868, 2.09523237694257881029268506584, 2.25905054736044839771886315695, 2.63127881566113622619271851729, 2.92942368087034259985950863045, 3.15027772003636097619838016139, 3.18160910696266606170493430349, 3.28757368619713100554309439841, 3.36422530565556940997342845762, 3.47584398137650241764381898772, 3.59712447432829192917643363262, 3.91926548229564893177850368276, 3.99304683954337036343731870272, 4.48105422986129119929578883854, 4.54047235218903439872826938531, 4.59039632014148456193568343942, 4.68845270593393339994967509620, 4.69505653065804084253686907256
Plot not available for L-functions of degree greater than 10.
