| L(s) = 1 | + 12·5-s − 16·9-s − 24·11-s + 68·19-s + 34·25-s − 192·45-s + 400·49-s − 288·55-s − 600·61-s + 80·81-s + 816·95-s + 384·99-s − 424·101-s − 940·121-s − 148·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 664·169-s − 1.08e3·171-s + 173-s + ⋯ |
| L(s) = 1 | + 12/5·5-s − 1.77·9-s − 2.18·11-s + 3.57·19-s + 1.35·25-s − 4.26·45-s + 8.16·49-s − 5.23·55-s − 9.83·61-s + 0.987·81-s + 8.58·95-s + 3.87·99-s − 4.19·101-s − 7.76·121-s − 1.18·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 3.92·169-s − 6.36·171-s + 0.00578·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{12} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{12} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.542026522\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.542026522\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 - 6 T + 37 T^{2} - 32 p T^{3} + 37 p^{2} T^{4} - 6 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 19 | \( ( 1 - 34 T + 723 T^{2} - 636 p T^{3} + 723 p^{2} T^{4} - 34 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| good | 3 | \( ( 1 + 8 T^{2} + 56 T^{4} - 110 T^{6} + 56 p^{4} T^{8} + 8 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 7 | \( ( 1 - 200 T^{2} + 19984 T^{4} - 24978 p^{2} T^{6} + 19984 p^{4} T^{8} - 200 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 11 | \( ( 1 + 6 T + 325 T^{2} + 128 p T^{3} + 325 p^{2} T^{4} + 6 p^{4} T^{5} + p^{6} T^{6} )^{4} \) |
| 13 | \( ( 1 + 332 T^{2} + 7272 p T^{4} + 15442282 T^{6} + 7272 p^{5} T^{8} + 332 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 17 | \( ( 1 - 1420 T^{2} + 900624 T^{4} - 331623882 T^{6} + 900624 p^{4} T^{8} - 1420 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 23 | \( ( 1 - 824 T^{2} + 720640 T^{4} - 450330930 T^{6} + 720640 p^{4} T^{8} - 824 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 29 | \( ( 1 - 2740 T^{2} + 4126128 T^{4} - 4996698 p^{2} T^{6} + 4126128 p^{4} T^{8} - 2740 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 31 | \( ( 1 - 1114 T^{2} + 641475 T^{4} + 395464236 T^{6} + 641475 p^{4} T^{8} - 1114 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 37 | \( ( 1 + 2902 T^{2} + 8216351 T^{4} + 11547943732 T^{6} + 8216351 p^{4} T^{8} + 2902 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 41 | \( ( 1 - 2946 T^{2} - 2428413 T^{4} + 15632004796 T^{6} - 2428413 p^{4} T^{8} - 2946 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 43 | \( ( 1 - 3554 T^{2} + 8177587 T^{4} - 18487447044 T^{6} + 8177587 p^{4} T^{8} - 3554 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 47 | \( ( 1 + 802 T^{2} + 9915583 T^{4} + 2414107836 T^{6} + 9915583 p^{4} T^{8} + 802 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 53 | \( ( 1 + 13724 T^{2} + 85490808 T^{4} + 307901289994 T^{6} + 85490808 p^{4} T^{8} + 13724 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 59 | \( ( 1 - 2800 T^{2} + 12497808 T^{4} - 96285230418 T^{6} + 12497808 p^{4} T^{8} - 2800 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 61 | \( ( 1 + 150 T + 16213 T^{2} + 1135264 T^{3} + 16213 p^{2} T^{4} + 150 p^{4} T^{5} + p^{6} T^{6} )^{4} \) |
| 67 | \( ( 1 + 4504 T^{2} + 47039432 T^{4} + 119843552146 T^{6} + 47039432 p^{4} T^{8} + 4504 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 71 | \( ( 1 - 18266 T^{2} + 170784163 T^{4} - 1026393192276 T^{6} + 170784163 p^{4} T^{8} - 18266 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 73 | \( ( 1 - 19260 T^{2} + 179069184 T^{4} - 1112885983082 T^{6} + 179069184 p^{4} T^{8} - 19260 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 79 | \( ( 1 - 11006 T^{2} + 39589567 T^{4} - 118680581124 T^{6} + 39589567 p^{4} T^{8} - 11006 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 83 | \( ( 1 - 21718 T^{2} + 291339135 T^{4} - 2389791397812 T^{6} + 291339135 p^{4} T^{8} - 21718 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 89 | \( ( 1 - 29018 T^{2} + 5189339 p T^{4} - 4481569968468 T^{6} + 5189339 p^{5} T^{8} - 29018 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 97 | \( ( 1 + 34446 T^{2} + 650399935 T^{4} + 7487480008036 T^{6} + 650399935 p^{4} T^{8} + 34446 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
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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
−3.54864247687809818463892347686, −3.37473713017714486696656205007, −3.21235741756006400021997413088, −3.18204277874404766501403457827, −2.89751063531027252024522482782, −2.87722239164198453667360650911, −2.71266351973069411779947267947, −2.70890878417706198517302896344, −2.65988820191810871636676367082, −2.58435283599740164008370843067, −2.45569087230764471771464694603, −2.44526121541778568478321366754, −2.25850608723946550506574558533, −2.23468901895107711676257622246, −1.71412966828098035244099118217, −1.58138793390594229243230355461, −1.45350913640438680696178142705, −1.41165369487116329093233338698, −1.34637246410947585287315476747, −1.29413158275924141739579991851, −1.18188198788380809185014597188, −0.64419538300605431407346247220, −0.47038436440563266581596509210, −0.32398908892569527003871447396, −0.10023122667104275332378500758,
0.10023122667104275332378500758, 0.32398908892569527003871447396, 0.47038436440563266581596509210, 0.64419538300605431407346247220, 1.18188198788380809185014597188, 1.29413158275924141739579991851, 1.34637246410947585287315476747, 1.41165369487116329093233338698, 1.45350913640438680696178142705, 1.58138793390594229243230355461, 1.71412966828098035244099118217, 2.23468901895107711676257622246, 2.25850608723946550506574558533, 2.44526121541778568478321366754, 2.45569087230764471771464694603, 2.58435283599740164008370843067, 2.65988820191810871636676367082, 2.70890878417706198517302896344, 2.71266351973069411779947267947, 2.87722239164198453667360650911, 2.89751063531027252024522482782, 3.18204277874404766501403457827, 3.21235741756006400021997413088, 3.37473713017714486696656205007, 3.54864247687809818463892347686
Plot not available for L-functions of degree greater than 10.
