| L(s) = 1 | − 60·5-s + 1.32e3·9-s − 6.41e3·25-s − 7.86e3·29-s + 6.55e4·41-s − 7.94e4·45-s + 1.09e5·49-s − 1.18e5·61-s + 7.73e5·81-s + 3.54e5·89-s − 4.90e5·101-s + 1.04e5·109-s − 6.23e5·121-s + 1.65e5·125-s + 127-s + 131-s + 137-s + 139-s + 4.71e5·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3.66e5·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 1.07·5-s + 5.44·9-s − 2.05·25-s − 1.73·29-s + 6.09·41-s − 5.84·45-s + 6.53·49-s − 4.06·61-s + 13.1·81-s + 4.74·89-s − 4.78·101-s + 0.845·109-s − 3.87·121-s + 0.946·125-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 1.86·145-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 0.987·169-s + 2.54e−6·173-s + 2.33e−6·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(22.20964463\) |
| \(L(\frac12)\) |
\(\approx\) |
\(22.20964463\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 + 6 p T + 911 p T^{2} + 10932 p^{2} T^{3} + 911 p^{6} T^{4} + 6 p^{11} T^{5} + p^{15} T^{6} )^{2} \) |
| good | 3 | \( ( 1 - 662 T^{2} + 10021 p^{3} T^{4} - 969236 p^{4} T^{6} + 10021 p^{13} T^{8} - 662 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 7 | \( ( 1 - 54910 T^{2} + 1199559327 T^{4} - 18786317442980 T^{6} + 1199559327 p^{10} T^{8} - 54910 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 11 | \( ( 1 + 311650 T^{2} + 88422679991 T^{4} + 16030422126529084 T^{6} + 88422679991 p^{10} T^{8} + 311650 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 13 | \( ( 1 - 183262 T^{2} + 260173900423 T^{4} - 19831011157232516 T^{6} + 260173900423 p^{10} T^{8} - 183262 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 17 | \( ( 1 + 68026 p T^{2} + 502148271983 T^{4} - 1980542974409409364 T^{6} + 502148271983 p^{10} T^{8} + 68026 p^{21} T^{10} + p^{30} T^{12} )^{2} \) |
| 19 | \( ( 1 + 3143250 T^{2} + 17341223849991 T^{4} + 36134402435803900316 T^{6} + 17341223849991 p^{10} T^{8} + 3143250 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 23 | \( ( 1 - 2071710 T^{2} + 29683977204927 T^{4} - \)\(44\!\cdots\!80\)\( T^{6} + 29683977204927 p^{10} T^{8} - 2071710 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 29 | \( ( 1 + 1966 T + 38511987 T^{2} + 51339089236 T^{3} + 38511987 p^{5} T^{4} + 1966 p^{10} T^{5} + p^{15} T^{6} )^{4} \) |
| 31 | \( ( 1 + 47574970 T^{2} + 2163796989249871 T^{4} + \)\(78\!\cdots\!04\)\( T^{6} + 2163796989249871 p^{10} T^{8} + 47574970 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 37 | \( ( 1 - 301201198 T^{2} + 42883739211630103 T^{4} - \)\(37\!\cdots\!24\)\( T^{6} + 42883739211630103 p^{10} T^{8} - 301201198 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 41 | \( ( 1 - 16390 T + 318388423 T^{2} - 3510238606580 T^{3} + 318388423 p^{5} T^{4} - 16390 p^{10} T^{5} + p^{15} T^{6} )^{4} \) |
| 43 | \( ( 1 - 640358182 T^{2} + 187647756282389367 T^{4} - \)\(33\!\cdots\!76\)\( T^{6} + 187647756282389367 p^{10} T^{8} - 640358182 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 47 | \( ( 1 - 764642190 T^{2} + 310900354010297967 T^{4} - \)\(85\!\cdots\!20\)\( T^{6} + 310900354010297967 p^{10} T^{8} - 764642190 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 53 | \( ( 1 - 1611835534 T^{2} + 1232324270520828407 T^{4} - \)\(61\!\cdots\!84\)\( T^{6} + 1232324270520828407 p^{10} T^{8} - 1611835534 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 59 | \( ( 1 + 2333761794 T^{2} + 3044701823751957015 T^{4} + \)\(26\!\cdots\!80\)\( T^{6} + 3044701823751957015 p^{10} T^{8} + 2333761794 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 61 | \( ( 1 + 29558 T + 2350852083 T^{2} + 48603849831556 T^{3} + 2350852083 p^{5} T^{4} + 29558 p^{10} T^{5} + p^{15} T^{6} )^{4} \) |
| 67 | \( ( 1 - 3741380758 T^{2} + 9572756841213379047 T^{4} - \)\(14\!\cdots\!84\)\( T^{6} + 9572756841213379047 p^{10} T^{8} - 3741380758 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 71 | \( ( 1 + 9793576042 T^{2} + 41626938509482688159 T^{4} + \)\(98\!\cdots\!36\)\( T^{6} + 41626938509482688159 p^{10} T^{8} + 9793576042 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 73 | \( ( 1 - 5477783158 T^{2} + 15527894807038343935 T^{4} - \)\(34\!\cdots\!40\)\( T^{6} + 15527894807038343935 p^{10} T^{8} - 5477783158 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 79 | \( ( 1 + 8854539610 T^{2} + 48426464203018517551 T^{4} + \)\(17\!\cdots\!76\)\( T^{6} + 48426464203018517551 p^{10} T^{8} + 8854539610 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 83 | \( ( 1 - 17886738486 T^{2} + \)\(14\!\cdots\!27\)\( T^{4} - \)\(73\!\cdots\!08\)\( T^{6} + \)\(14\!\cdots\!27\)\( p^{10} T^{8} - 17886738486 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 89 | \( ( 1 - 88734 T + 11682221271 T^{2} - 581994460142148 T^{3} + 11682221271 p^{5} T^{4} - 88734 p^{10} T^{5} + p^{15} T^{6} )^{4} \) |
| 97 | \( ( 1 - 22620660742 T^{2} + \)\(25\!\cdots\!35\)\( T^{4} - \)\(22\!\cdots\!60\)\( T^{6} + \)\(25\!\cdots\!35\)\( p^{10} T^{8} - 22620660742 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.63646593284941763354410632784, −3.59763246624247033366688291669, −3.19717713160038653033053767317, −3.07157707416393067235942446584, −3.01446415640144602408057534159, −2.82650240861355470261679046819, −2.78394668976824701199523849028, −2.49738169284544327248127053745, −2.34908523879968849398026466432, −2.27136148484139237088018510322, −2.27047464197409627081097680389, −2.05727502674387213756147960951, −1.88971198984913344762289328174, −1.65181339006130151791092158244, −1.51679906694415746404612464742, −1.46821256493303729458895127737, −1.35740480538044642471475506766, −1.30053713468178110137328062222, −1.07361257774170848939292074699, −0.870451268312558138456742924239, −0.67249609560643556372348706082, −0.58149714460924359751501040320, −0.50873939816071449862987491288, −0.44340113818074821204598398380, −0.12383073429486906305957852043,
0.12383073429486906305957852043, 0.44340113818074821204598398380, 0.50873939816071449862987491288, 0.58149714460924359751501040320, 0.67249609560643556372348706082, 0.870451268312558138456742924239, 1.07361257774170848939292074699, 1.30053713468178110137328062222, 1.35740480538044642471475506766, 1.46821256493303729458895127737, 1.51679906694415746404612464742, 1.65181339006130151791092158244, 1.88971198984913344762289328174, 2.05727502674387213756147960951, 2.27047464197409627081097680389, 2.27136148484139237088018510322, 2.34908523879968849398026466432, 2.49738169284544327248127053745, 2.78394668976824701199523849028, 2.82650240861355470261679046819, 3.01446415640144602408057534159, 3.07157707416393067235942446584, 3.19717713160038653033053767317, 3.59763246624247033366688291669, 3.63646593284941763354410632784
Plot not available for L-functions of degree greater than 10.
