| L(s) = 1 | + 2·4-s − 5·9-s + 18·11-s + 16-s + 24·19-s + 18·31-s − 10·36-s + 36·41-s + 36·44-s − 22·49-s + 18·59-s − 54·61-s − 2·64-s + 48·76-s + 2·79-s + 9·81-s − 6·89-s − 90·99-s + 54·101-s + 2·109-s + 137·121-s + 36·124-s + 127-s + 131-s + 137-s + 139-s − 5·144-s + ⋯ |
| L(s) = 1 | + 4-s − 5/3·9-s + 5.42·11-s + 1/4·16-s + 5.50·19-s + 3.23·31-s − 5/3·36-s + 5.62·41-s + 5.42·44-s − 3.14·49-s + 2.34·59-s − 6.91·61-s − 1/4·64-s + 5.50·76-s + 0.225·79-s + 81-s − 0.635·89-s − 9.04·99-s + 5.37·101-s + 0.191·109-s + 12.4·121-s + 3.23·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.416·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(36.85794853\) |
| \(L(\frac12)\) |
\(\approx\) |
\(36.85794853\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 3 | \( 1 + 5 T^{2} + 16 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | \( 1 \) |
| 7 | \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 11 | \( ( 1 - 9 T + 53 T^{2} - 234 T^{3} + 852 T^{4} - 234 p T^{5} + 53 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 + 10 T^{2} - 189 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 7 T + p T^{2} )^{4}( 1 + T + p T^{2} )^{4} \) |
| 23 | \( 1 + 71 T^{2} + 2797 T^{4} + 84206 T^{6} + 2089006 T^{8} + 84206 p^{2} T^{10} + 2797 p^{4} T^{12} + 71 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 - 47 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 9 T + 71 T^{2} - 396 T^{3} + 1812 T^{4} - 396 p T^{5} + 71 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( 1 + 56 T^{2} + 802 T^{4} - 22624 T^{6} - 719789 T^{8} - 22624 p^{2} T^{10} + 802 p^{4} T^{12} + 56 p^{6} T^{14} + p^{8} T^{16} \) |
| 41 | \( ( 1 - 9 T + 94 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 + 49 T^{2} + 660 T^{4} + 49 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 - 112 T^{2} + 6178 T^{4} - 218176 T^{6} + 7936579 T^{8} - 218176 p^{2} T^{10} + 6178 p^{4} T^{12} - 112 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( 1 + 191 T^{2} + 21817 T^{4} + 1727786 T^{6} + 104667286 T^{8} + 1727786 p^{2} T^{10} + 21817 p^{4} T^{12} + 191 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( ( 1 - 9 T + 17 T^{2} + 486 T^{3} - 4164 T^{4} + 486 p T^{5} + 17 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 27 T + 401 T^{2} + 4266 T^{3} + 36066 T^{4} + 4266 p T^{5} + 401 p^{2} T^{6} + 27 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 - 205 T^{2} + 23209 T^{4} - 2016790 T^{6} + 145240510 T^{8} - 2016790 p^{2} T^{10} + 23209 p^{4} T^{12} - 205 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 208 T^{2} + 19710 T^{4} - 208 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 142 T^{2} + 14835 T^{4} + 142 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - T - 83 T^{2} + 74 T^{3} + 736 T^{4} + 74 p T^{5} - 83 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 190 T^{2} + 18051 T^{4} + 190 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 3 T - 163 T^{2} - 18 T^{3} + 20862 T^{4} - 18 p T^{5} - 163 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 155 T^{2} + 12276 T^{4} - 155 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−4.12457992854145188086290790745, −4.06564665927553749122160827541, −4.03906740368883889747964011234, −4.00709064604060904336846632044, −3.57167440562203668200974417473, −3.51286394885295589113720397164, −3.28289841177660882966997643220, −3.23773255808426764563421952153, −3.22900126359394728051601663999, −3.11964157602731953243678227444, −2.89432053665106211011601657874, −2.88034561405029184213628923870, −2.84033802297341403846624376642, −2.31425800753544516117366086696, −2.30625785542197250308658583070, −2.27364949908831580316097884125, −1.82439431673180624317097021825, −1.68111627125191654000801435763, −1.46033874706943777590915825294, −1.42039078835561181707089364489, −1.28043573366213454695672427824, −0.937234226211828056839492985000, −0.932530888172830631748354960670, −0.71891766425160026988386325819, −0.58721760981048303293465857233,
0.58721760981048303293465857233, 0.71891766425160026988386325819, 0.932530888172830631748354960670, 0.937234226211828056839492985000, 1.28043573366213454695672427824, 1.42039078835561181707089364489, 1.46033874706943777590915825294, 1.68111627125191654000801435763, 1.82439431673180624317097021825, 2.27364949908831580316097884125, 2.30625785542197250308658583070, 2.31425800753544516117366086696, 2.84033802297341403846624376642, 2.88034561405029184213628923870, 2.89432053665106211011601657874, 3.11964157602731953243678227444, 3.22900126359394728051601663999, 3.23773255808426764563421952153, 3.28289841177660882966997643220, 3.51286394885295589113720397164, 3.57167440562203668200974417473, 4.00709064604060904336846632044, 4.03906740368883889747964011234, 4.06564665927553749122160827541, 4.12457992854145188086290790745
Plot not available for L-functions of degree greater than 10.
