| L(s) = 1 | − 2·3-s + 46·4-s + 34·9-s − 92·12-s + 420·13-s + 803·16-s − 372·19-s + 3.02e3·25-s − 686·27-s − 2.77e3·31-s + 1.56e3·36-s − 2.56e3·37-s − 840·39-s + 4.72e3·43-s − 1.60e3·48-s + 1.37e3·49-s + 1.93e4·52-s + 744·57-s + 972·61-s + 7.10e3·64-s + 1.02e4·67-s − 3.20e4·73-s − 6.05e3·75-s − 1.71e4·76-s − 2.31e4·79-s − 2.48e3·81-s + 5.55e3·93-s + ⋯ |
| L(s) = 1 | − 2/9·3-s + 23/8·4-s + 0.419·9-s − 0.638·12-s + 2.48·13-s + 3.13·16-s − 1.03·19-s + 4.84·25-s − 0.941·27-s − 2.88·31-s + 1.20·36-s − 1.86·37-s − 0.552·39-s + 2.55·43-s − 0.697·48-s + 4/7·49-s + 7.14·52-s + 0.228·57-s + 0.261·61-s + 1.73·64-s + 2.27·67-s − 6.00·73-s − 1.07·75-s − 2.96·76-s − 3.71·79-s − 0.378·81-s + 0.641·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(7.002652827\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.002652827\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 2 T - 10 p T^{2} + 62 p^{2} T^{3} + 74 p^{4} T^{4} + 62 p^{6} T^{5} - 10 p^{9} T^{6} + 2 p^{12} T^{7} + p^{16} T^{8} \) |
| 7 | \( ( 1 - p^{3} T^{2} )^{4} \) |
| good | 2 | \( 1 - 23 p T^{2} + 1313 T^{4} - 7641 p^{2} T^{6} + 138077 p^{2} T^{8} - 7641 p^{10} T^{10} + 1313 p^{16} T^{12} - 23 p^{25} T^{14} + p^{32} T^{16} \) |
| 5 | \( 1 - 3028 T^{2} + 4849436 T^{4} - 5035499772 T^{6} + 3711566600246 T^{8} - 5035499772 p^{8} T^{10} + 4849436 p^{16} T^{12} - 3028 p^{24} T^{14} + p^{32} T^{16} \) |
| 11 | \( 1 - 27532 T^{2} + 793925288 T^{4} - 14029736896548 T^{6} + 258697335125368526 T^{8} - 14029736896548 p^{8} T^{10} + 793925288 p^{16} T^{12} - 27532 p^{24} T^{14} + p^{32} T^{16} \) |
| 13 | \( ( 1 - 210 T + 63290 T^{2} - 9829302 T^{3} + 2121485946 T^{4} - 9829302 p^{4} T^{5} + 63290 p^{8} T^{6} - 210 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 17 | \( 1 - 348356 T^{2} + 64048177720 T^{4} - 8167237508535212 T^{6} + \)\(78\!\cdots\!50\)\( T^{8} - 8167237508535212 p^{8} T^{10} + 64048177720 p^{16} T^{12} - 348356 p^{24} T^{14} + p^{32} T^{16} \) |
| 19 | \( ( 1 + 186 T + 342782 T^{2} + 3334458 p T^{3} + 61206051834 T^{4} + 3334458 p^{5} T^{5} + 342782 p^{8} T^{6} + 186 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 23 | \( 1 - 1111940 T^{2} + 636070185160 T^{4} - 259661986150766348 T^{6} + \)\(82\!\cdots\!90\)\( T^{8} - 259661986150766348 p^{8} T^{10} + 636070185160 p^{16} T^{12} - 1111940 p^{24} T^{14} + p^{32} T^{16} \) |
| 29 | \( 1 - 653584 T^{2} + 1023644128220 T^{4} - 7886970213722544 p T^{6} + \)\(47\!\cdots\!62\)\( T^{8} - 7886970213722544 p^{9} T^{10} + 1023644128220 p^{16} T^{12} - 653584 p^{24} T^{14} + p^{32} T^{16} \) |
| 31 | \( ( 1 + 1388 T + 3392912 T^{2} + 3467620980 T^{3} + 4635546858686 T^{4} + 3467620980 p^{4} T^{5} + 3392912 p^{8} T^{6} + 1388 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 37 | \( ( 1 + 1280 T + 5575880 T^{2} + 5329274592 T^{3} + 13555511794190 T^{4} + 5329274592 p^{4} T^{5} + 5575880 p^{8} T^{6} + 1280 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 41 | \( 1 - 11989972 T^{2} + 64687127613080 T^{4} - \)\(22\!\cdots\!56\)\( T^{6} + \)\(64\!\cdots\!50\)\( T^{8} - \)\(22\!\cdots\!56\)\( p^{8} T^{10} + 64687127613080 p^{16} T^{12} - 11989972 p^{24} T^{14} + p^{32} T^{16} \) |
| 43 | \( ( 1 - 2360 T + 9057832 T^{2} - 24907997576 T^{3} + 38674546313998 T^{4} - 24907997576 p^{4} T^{5} + 9057832 p^{8} T^{6} - 2360 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 47 | \( 1 - 5336888 T^{2} + 55522526758780 T^{4} - 8072213151432963448 p T^{6} + \)\(15\!\cdots\!82\)\( T^{8} - 8072213151432963448 p^{9} T^{10} + 55522526758780 p^{16} T^{12} - 5336888 p^{24} T^{14} + p^{32} T^{16} \) |
| 53 | \( 1 - 36244496 T^{2} + 649309191943708 T^{4} - \)\(76\!\cdots\!04\)\( T^{6} + \)\(68\!\cdots\!98\)\( T^{8} - \)\(76\!\cdots\!04\)\( p^{8} T^{10} + 649309191943708 p^{16} T^{12} - 36244496 p^{24} T^{14} + p^{32} T^{16} \) |
| 59 | \( 1 - 1467520 p T^{2} + 3375884924194624 T^{4} - \)\(77\!\cdots\!08\)\( T^{6} + \)\(11\!\cdots\!30\)\( T^{8} - \)\(77\!\cdots\!08\)\( p^{8} T^{10} + 3375884924194624 p^{16} T^{12} - 1467520 p^{25} T^{14} + p^{32} T^{16} \) |
| 61 | \( ( 1 - 486 T + 44977274 T^{2} - 12974067234 T^{3} + 875583356937594 T^{4} - 12974067234 p^{4} T^{5} + 44977274 p^{8} T^{6} - 486 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 67 | \( ( 1 - 5100 T + 56441480 T^{2} - 236150562468 T^{3} + 1479452251001166 T^{4} - 236150562468 p^{4} T^{5} + 56441480 p^{8} T^{6} - 5100 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 71 | \( 1 - 151515260 T^{2} + 11001738004368328 T^{4} - \)\(49\!\cdots\!92\)\( T^{6} + \)\(15\!\cdots\!10\)\( T^{8} - \)\(49\!\cdots\!92\)\( p^{8} T^{10} + 11001738004368328 p^{16} T^{12} - 151515260 p^{24} T^{14} + p^{32} T^{16} \) |
| 73 | \( ( 1 + 16004 T + 162466988 T^{2} + 1171637909916 T^{3} + 6877062230867558 T^{4} + 1171637909916 p^{4} T^{5} + 162466988 p^{8} T^{6} + 16004 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 79 | \( ( 1 + 11584 T + 154069540 T^{2} + 1153659898816 T^{3} + 9268831383054406 T^{4} + 1153659898816 p^{4} T^{5} + 154069540 p^{8} T^{6} + 11584 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 83 | \( 1 - 241219280 T^{2} + 28240892356817056 T^{4} - \)\(21\!\cdots\!68\)\( T^{6} + \)\(11\!\cdots\!26\)\( T^{8} - \)\(21\!\cdots\!68\)\( p^{8} T^{10} + 28240892356817056 p^{16} T^{12} - 241219280 p^{24} T^{14} + p^{32} T^{16} \) |
| 89 | \( 1 - 401576116 T^{2} + 75457229541260312 T^{4} - \)\(86\!\cdots\!68\)\( T^{6} + \)\(65\!\cdots\!38\)\( T^{8} - \)\(86\!\cdots\!68\)\( p^{8} T^{10} + 75457229541260312 p^{16} T^{12} - 401576116 p^{24} T^{14} + p^{32} T^{16} \) |
| 97 | \( ( 1 - 14056 T + 275661116 T^{2} - 2652268176600 T^{3} + 30780781701641222 T^{4} - 2652268176600 p^{4} T^{5} + 275661116 p^{8} T^{6} - 14056 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
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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
−8.425570021597585141457832875865, −8.377534191316208802675623087843, −7.39341086560479477182871601254, −7.37318528442084879922913967975, −7.35422137938946604432712653396, −7.33413563077518706760758789798, −6.99704922761702684699745317596, −6.86614255847059640845768959461, −6.57190549510020837946762670910, −6.20876360083180183931966219730, −6.00177983993118042301691130747, −5.93813158583256962790175183312, −5.78590502111366779405330595128, −5.29310007023633282812946220279, −4.99460734693441660767760364636, −4.48012829750994238842254122008, −4.41900249527205913815427093213, −3.84026594915976863717106959289, −3.50295475604614218864400523024, −3.18218485444991793272053194447, −2.82244452866457362295307980017, −2.40600230391926213365845453536, −1.80301479030275643572363310825, −1.60753900056563271911885381433, −0.885770348534765917784617129662,
0.885770348534765917784617129662, 1.60753900056563271911885381433, 1.80301479030275643572363310825, 2.40600230391926213365845453536, 2.82244452866457362295307980017, 3.18218485444991793272053194447, 3.50295475604614218864400523024, 3.84026594915976863717106959289, 4.41900249527205913815427093213, 4.48012829750994238842254122008, 4.99460734693441660767760364636, 5.29310007023633282812946220279, 5.78590502111366779405330595128, 5.93813158583256962790175183312, 6.00177983993118042301691130747, 6.20876360083180183931966219730, 6.57190549510020837946762670910, 6.86614255847059640845768959461, 6.99704922761702684699745317596, 7.33413563077518706760758789798, 7.35422137938946604432712653396, 7.37318528442084879922913967975, 7.39341086560479477182871601254, 8.377534191316208802675623087843, 8.425570021597585141457832875865
Plot not available for L-functions of degree greater than 10.
