| L(s) = 1 | − 32·4-s + 3.39e3·9-s − 4.72e3·16-s + 1.60e4·29-s − 1.08e5·36-s − 1.92e5·41-s + 6.32e5·49-s + 2.15e5·61-s + 1.63e5·64-s + 4.82e6·81-s + 4.34e6·89-s + 8.10e6·101-s − 2.56e6·109-s − 5.14e5·116-s + 8.46e6·121-s + 127-s + 131-s + 137-s + 139-s − 1.60e7·144-s + 149-s + 151-s + 157-s + 163-s + 6.14e6·164-s + 167-s − 1.32e7·169-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 4.65·9-s − 1.15·16-s + 0.658·29-s − 2.32·36-s − 2.78·41-s + 5.38·49-s + 0.948·61-s + 0.622·64-s + 9.08·81-s + 6.16·89-s + 7.87·101-s − 1.98·109-s − 0.329·116-s + 4.77·121-s − 5.36·144-s + 1.39·164-s − 2.74·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+3)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(42.08534882\) |
| \(L(\frac12)\) |
\(\approx\) |
\(42.08534882\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{5} T^{2} + 359 p^{4} T^{4} + 671 p^{8} T^{6} + 359 p^{16} T^{8} + p^{29} T^{10} + p^{36} T^{12} \) |
| 5 | \( 1 \) |
| good | 3 | \( ( 1 - 566 p T^{2} + 212311 p^{2} T^{4} - 6975508 p^{5} T^{6} + 212311 p^{14} T^{8} - 566 p^{25} T^{10} + p^{36} T^{12} )^{2} \) |
| 7 | \( ( 1 - 45214 p T^{2} + 63613951199 T^{4} - 187302694224796 p^{2} T^{6} + 63613951199 p^{12} T^{8} - 45214 p^{25} T^{10} + p^{36} T^{12} )^{2} \) |
| 11 | \( ( 1 - 4230966 T^{2} + 12435345552255 T^{4} - 27133599137440282100 T^{6} + 12435345552255 p^{12} T^{8} - 4230966 p^{24} T^{10} + p^{36} T^{12} )^{2} \) |
| 13 | \( ( 1 + 6612822 T^{2} + 51291900208479 T^{4} + \)\(31\!\cdots\!16\)\( T^{6} + 51291900208479 p^{12} T^{8} + 6612822 p^{24} T^{10} + p^{36} T^{12} )^{2} \) |
| 17 | \( ( 1 + 80718662 T^{2} + 3795004300453199 T^{4} + \)\(11\!\cdots\!16\)\( T^{6} + 3795004300453199 p^{12} T^{8} + 80718662 p^{24} T^{10} + p^{36} T^{12} )^{2} \) |
| 19 | \( ( 1 - 151791126 T^{2} + 12887986688168415 T^{4} - \)\(73\!\cdots\!40\)\( T^{6} + 12887986688168415 p^{12} T^{8} - 151791126 p^{24} T^{10} + p^{36} T^{12} )^{2} \) |
| 23 | \( ( 1 - 838151698 T^{2} + 299619665559292319 T^{4} - \)\(58\!\cdots\!64\)\( T^{6} + 299619665559292319 p^{12} T^{8} - 838151698 p^{24} T^{10} + p^{36} T^{12} )^{2} \) |
| 29 | \( ( 1 - 4018 T + 984038839 T^{2} + 4482562456036 T^{3} + 984038839 p^{6} T^{4} - 4018 p^{12} T^{5} + p^{18} T^{6} )^{4} \) |
| 31 | \( ( 1 - 34825146 p T^{2} + 1559430337965859215 T^{4} - \)\(97\!\cdots\!40\)\( T^{6} + 1559430337965859215 p^{12} T^{8} - 34825146 p^{25} T^{10} + p^{36} T^{12} )^{2} \) |
| 37 | \( ( 1 + 14610185142 T^{2} + 90894099761693350719 T^{4} + \)\(30\!\cdots\!96\)\( T^{6} + 90894099761693350719 p^{12} T^{8} + 14610185142 p^{24} T^{10} + p^{36} T^{12} )^{2} \) |
| 41 | \( ( 1 + 48034 T + 14394061135 T^{2} + 453005557569020 T^{3} + 14394061135 p^{6} T^{4} + 48034 p^{12} T^{5} + p^{18} T^{6} )^{4} \) |
| 43 | \( ( 1 - 17834866498 T^{2} + \)\(21\!\cdots\!99\)\( T^{4} - \)\(15\!\cdots\!04\)\( T^{6} + \)\(21\!\cdots\!99\)\( p^{12} T^{8} - 17834866498 p^{24} T^{10} + p^{36} T^{12} )^{2} \) |
| 47 | \( ( 1 - 27982006258 T^{2} + \)\(51\!\cdots\!99\)\( T^{4} - \)\(67\!\cdots\!24\)\( T^{6} + \)\(51\!\cdots\!99\)\( p^{12} T^{8} - 27982006258 p^{24} T^{10} + p^{36} T^{12} )^{2} \) |
| 53 | \( ( 1 + 53634231542 T^{2} + \)\(53\!\cdots\!59\)\( T^{4} - \)\(74\!\cdots\!44\)\( T^{6} + \)\(53\!\cdots\!59\)\( p^{12} T^{8} + 53634231542 p^{24} T^{10} + p^{36} T^{12} )^{2} \) |
| 59 | \( ( 1 - 146094395766 T^{2} + \)\(10\!\cdots\!35\)\( T^{4} - \)\(14\!\cdots\!20\)\( p^{2} T^{6} + \)\(10\!\cdots\!35\)\( p^{12} T^{8} - 146094395766 p^{24} T^{10} + p^{36} T^{12} )^{2} \) |
| 61 | \( ( 1 - 53846 T + 87983744615 T^{2} - 7715289154258100 T^{3} + 87983744615 p^{6} T^{4} - 53846 p^{12} T^{5} + p^{18} T^{6} )^{4} \) |
| 67 | \( ( 1 - 283076108578 T^{2} + \)\(40\!\cdots\!79\)\( T^{4} - \)\(40\!\cdots\!64\)\( T^{6} + \)\(40\!\cdots\!79\)\( p^{12} T^{8} - 283076108578 p^{24} T^{10} + p^{36} T^{12} )^{2} \) |
| 71 | \( ( 1 - 612196656486 T^{2} + \)\(17\!\cdots\!95\)\( T^{4} - \)\(28\!\cdots\!60\)\( T^{6} + \)\(17\!\cdots\!95\)\( p^{12} T^{8} - 612196656486 p^{24} T^{10} + p^{36} T^{12} )^{2} \) |
| 73 | \( ( 1 + 278917873062 T^{2} + \)\(54\!\cdots\!39\)\( T^{4} + \)\(10\!\cdots\!76\)\( T^{6} + \)\(54\!\cdots\!39\)\( p^{12} T^{8} + 278917873062 p^{24} T^{10} + p^{36} T^{12} )^{2} \) |
| 79 | \( ( 1 - 833901716166 T^{2} + \)\(35\!\cdots\!15\)\( T^{4} - \)\(10\!\cdots\!40\)\( T^{6} + \)\(35\!\cdots\!15\)\( p^{12} T^{8} - 833901716166 p^{24} T^{10} + p^{36} T^{12} )^{2} \) |
| 83 | \( ( 1 - 772762953058 T^{2} + \)\(16\!\cdots\!79\)\( T^{4} - \)\(14\!\cdots\!04\)\( T^{6} + \)\(16\!\cdots\!79\)\( p^{12} T^{8} - 772762953058 p^{24} T^{10} + p^{36} T^{12} )^{2} \) |
| 89 | \( ( 1 - 1086538 T + 1626117919039 T^{2} - 1075366431700477964 T^{3} + 1626117919039 p^{6} T^{4} - 1086538 p^{12} T^{5} + p^{18} T^{6} )^{4} \) |
| 97 | \( ( 1 + 2704280608902 T^{2} + \)\(40\!\cdots\!19\)\( T^{4} + \)\(41\!\cdots\!16\)\( T^{6} + \)\(40\!\cdots\!19\)\( p^{12} T^{8} + 2704280608902 p^{24} T^{10} + p^{36} T^{12} )^{2} \) |
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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
−3.70596395340336784902598165035, −3.63695089207543363230535656540, −3.49721507814831115932897280641, −3.47997516800511309344845473044, −3.27558545617689061760183540256, −3.12617598839640290731992077385, −2.67145852498795197271230384119, −2.61563280651221345894516726347, −2.58944831211210272800062403595, −2.34837160322012016605740283866, −2.31438400082318831980973010401, −1.97668504348823879377656960445, −1.97043267686290659937584668757, −1.87856577276333321216125698389, −1.69687722647776218921519975472, −1.67118810278021424800982877861, −1.33752778500833015433176145609, −1.17763395472579786886697933419, −0.942746391031468824158838242629, −0.941276627185301575415318407294, −0.902693942768918303000199691805, −0.65162722073464092104035021661, −0.43552188807879580560674953436, −0.38628922069693622876914587596, −0.21290559019169726914758059680,
0.21290559019169726914758059680, 0.38628922069693622876914587596, 0.43552188807879580560674953436, 0.65162722073464092104035021661, 0.902693942768918303000199691805, 0.941276627185301575415318407294, 0.942746391031468824158838242629, 1.17763395472579786886697933419, 1.33752778500833015433176145609, 1.67118810278021424800982877861, 1.69687722647776218921519975472, 1.87856577276333321216125698389, 1.97043267686290659937584668757, 1.97668504348823879377656960445, 2.31438400082318831980973010401, 2.34837160322012016605740283866, 2.58944831211210272800062403595, 2.61563280651221345894516726347, 2.67145852498795197271230384119, 3.12617598839640290731992077385, 3.27558545617689061760183540256, 3.47997516800511309344845473044, 3.49721507814831115932897280641, 3.63695089207543363230535656540, 3.70596395340336784902598165035
Plot not available for L-functions of degree greater than 10.
