| L(s) = 1 | + 4·4-s + 12·11-s + 9·16-s + 8·19-s + 12·29-s + 4·31-s + 24·41-s + 48·44-s + 16·49-s + 24·59-s − 8·61-s + 12·64-s − 72·71-s + 32·76-s − 16·79-s − 24·89-s + 24·101-s + 32·109-s + 48·116-s + 92·121-s + 16·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 2·4-s + 3.61·11-s + 9/4·16-s + 1.83·19-s + 2.22·29-s + 0.718·31-s + 3.74·41-s + 7.23·44-s + 16/7·49-s + 3.12·59-s − 1.02·61-s + 3/2·64-s − 8.54·71-s + 3.67·76-s − 1.80·79-s − 2.54·89-s + 2.38·101-s + 3.06·109-s + 4.45·116-s + 8.36·121-s + 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(24.79570606\) |
| \(L(\frac12)\) |
\(\approx\) |
\(24.79570606\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - p^{2} T^{2} + 7 T^{4} - p^{2} T^{6} + T^{8} - p^{4} T^{10} + 7 p^{4} T^{12} - p^{8} T^{14} + p^{8} T^{16} \) |
| 7 | \( 1 - 16 T^{2} + 121 T^{4} - 592 T^{6} + 3280 T^{8} - 592 p^{2} T^{10} + 121 p^{4} T^{12} - 16 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( ( 1 - 6 T + 8 T^{2} - 36 T^{3} + 267 T^{4} - 36 p T^{5} + 8 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 20 T^{2} + 231 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 40 T^{2} + 786 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 2 T + 36 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 23 | \( 1 - 88 T^{2} + 4753 T^{4} - 170104 T^{6} + 4569664 T^{8} - 170104 p^{2} T^{10} + 4753 p^{4} T^{12} - 88 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 - 6 T - 19 T^{2} + 18 T^{3} + 1140 T^{4} + 18 p T^{5} - 19 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 2 T - 56 T^{2} + 4 T^{3} + 2515 T^{4} + 4 p T^{5} - 56 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 56 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 12 T + 29 T^{2} - 396 T^{3} + 5640 T^{4} - 396 p T^{5} + 29 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 - 88 T^{2} + 3838 T^{4} - 18304 T^{6} - 2480621 T^{8} - 18304 p^{2} T^{10} + 3838 p^{4} T^{12} - 88 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( 1 - 160 T^{2} + 14809 T^{4} - 1019680 T^{6} + 55015600 T^{8} - 1019680 p^{2} T^{10} + 14809 p^{4} T^{12} - 160 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( ( 1 + 196 T^{2} + 15174 T^{4} + 196 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 12 T + 2 T^{2} - 288 T^{3} + 8187 T^{4} - 288 p T^{5} + 2 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 4 T - 35 T^{2} - 284 T^{3} - 2096 T^{4} - 284 p T^{5} - 35 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 - 184 T^{2} + 16657 T^{4} - 1512664 T^{6} + 123674896 T^{8} - 1512664 p^{2} T^{10} + 16657 p^{4} T^{12} - 184 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( ( 1 + 18 T + 196 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + 8 T - 98 T^{2} + 32 T^{3} + 14947 T^{4} + 32 p T^{5} - 98 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 160 T^{2} + 5929 T^{4} - 11360 p T^{6} + 23680 p^{2} T^{8} - 11360 p^{3} T^{10} + 5929 p^{4} T^{12} - 160 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( ( 1 + 6 T + 79 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 97 | \( 1 - 52 T^{2} - 12902 T^{4} + 167024 T^{6} + 129576019 T^{8} + 167024 p^{2} T^{10} - 12902 p^{4} T^{12} - 52 p^{6} T^{14} + p^{8} T^{16} \) |
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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
−4.48830709683416281185214010457, −4.41667318973970983146238965452, −4.31531077964427391843823362772, −4.30126455076371145414959754040, −4.29670226883442904491842142145, −3.79464600876999890857437818036, −3.64039194277109783683069256531, −3.38379453080188592334922313996, −3.37758279118044611017940782397, −3.36274052049625360057047830723, −3.16068261086876051483796408053, −3.05808410808527275324793207420, −2.80613856604271623709203590468, −2.62059700382854803220302078075, −2.59421609424629520413451828112, −2.28709588926291308269007573554, −2.18320602064042363591302582419, −2.02031537197286078017847787873, −1.78709758585820404782830364033, −1.53242947596527587557465361742, −1.25675654337675002154113818526, −1.11625810693004984083658261433, −1.07305219737098340952357839981, −0.977560419106518621726127019490, −0.53051108341858227404298957611,
0.53051108341858227404298957611, 0.977560419106518621726127019490, 1.07305219737098340952357839981, 1.11625810693004984083658261433, 1.25675654337675002154113818526, 1.53242947596527587557465361742, 1.78709758585820404782830364033, 2.02031537197286078017847787873, 2.18320602064042363591302582419, 2.28709588926291308269007573554, 2.59421609424629520413451828112, 2.62059700382854803220302078075, 2.80613856604271623709203590468, 3.05808410808527275324793207420, 3.16068261086876051483796408053, 3.36274052049625360057047830723, 3.37758279118044611017940782397, 3.38379453080188592334922313996, 3.64039194277109783683069256531, 3.79464600876999890857437818036, 4.29670226883442904491842142145, 4.30126455076371145414959754040, 4.31531077964427391843823362772, 4.41667318973970983146238965452, 4.48830709683416281185214010457
Plot not available for L-functions of degree greater than 10.
