| L(s) = 1 | + 24·5-s − 296·13-s + 600·17-s − 476·25-s + 888·29-s + 4.40e3·37-s − 552·41-s + 4.51e3·49-s + 5.11e3·53-s − 4.23e3·61-s − 7.10e3·65-s + 8.84e3·73-s + 1.44e4·85-s + 2.50e4·89-s + 2.30e4·97-s + 2.77e4·101-s + 3.52e4·109-s + 1.33e4·113-s + 3.61e4·121-s − 1.07e4·125-s + 127-s + 131-s + 137-s + 139-s + 2.13e4·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 0.959·5-s − 1.75·13-s + 2.07·17-s − 0.761·25-s + 1.05·29-s + 3.21·37-s − 0.328·41-s + 1.88·49-s + 1.81·53-s − 1.13·61-s − 1.68·65-s + 1.65·73-s + 1.99·85-s + 3.16·89-s + 2.44·97-s + 2.72·101-s + 2.96·109-s + 1.04·113-s + 2.47·121-s − 0.689·125-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 1.01·145-s + 4.50e−5·149-s + 4.38e−5·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(13.17000591\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.17000591\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( ( 1 - 12 T + 454 T^{2} - 12 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 4516 T^{2} + 16148934 T^{4} - 4516 p^{8} T^{6} + p^{16} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 36196 T^{2} + 708332166 T^{4} - 36196 p^{8} T^{6} + p^{16} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 148 T + 32646 T^{2} + 148 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 - 300 T + 186214 T^{2} - 300 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 195556 T^{2} + 17286835974 T^{4} - 195556 p^{8} T^{6} + p^{16} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 655492 T^{2} + 482373414 p^{2} T^{4} - 655492 p^{8} T^{6} + p^{16} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 444 T + 1423078 T^{2} - 444 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 67420 p T^{2} + 2465294161734 T^{4} - 67420 p^{9} T^{6} + p^{16} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 2204 T + 4842918 T^{2} - 2204 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 276 T + 5507494 T^{2} + 276 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 3593572 T^{2} + 5893643026566 T^{4} - 3593572 p^{8} T^{6} + p^{16} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 16853380 T^{2} + 118578854811654 T^{4} - 16853380 p^{8} T^{6} + p^{16} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 2556 T + 17173798 T^{2} - 2556 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 32722276 T^{2} + 559903452302214 T^{4} - 32722276 p^{8} T^{6} + p^{16} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 2116 T + 16830246 T^{2} + 2116 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 53256676 T^{2} + 1338152481850374 T^{4} - 53256676 p^{8} T^{6} + p^{16} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 79127428 T^{2} + 2740885222137990 T^{4} - 79127428 p^{8} T^{6} + p^{16} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 4420 T + 3693510 T^{2} - 4420 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 86136100 T^{2} + 4459690111983174 T^{4} - 86136100 p^{8} T^{6} + p^{16} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 855268 T^{2} + 4298268871421190 T^{4} - 855268 p^{8} T^{6} + p^{16} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 12540 T + 132835270 T^{2} - 12540 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 11524 T + 146880774 T^{2} - 11524 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.39000819558517722275022274465, −6.92234774100007911147883693007, −6.47533439633208189711279216095, −6.37719540197043622155254838869, −6.06155203519871845593758221394, −5.85521341366865022521674848170, −5.67566761043521053847052185471, −5.48242410780587910399815249741, −5.30735160576390507043352340233, −4.69665151853057699091628749011, −4.54531782365560943770474786961, −4.53189417817709244963502839432, −4.35984191602282888865110171759, −3.61833418753465841126182686572, −3.35891943167020866143276455061, −3.25989826073943185683030017111, −3.08547771765518802723362042798, −2.35449120286513803042115668373, −2.20953140216932312986264043543, −2.12178002997223937500318083580, −1.96165698914106258090616703328, −1.04098466708431820724940601665, −0.948981495105276082949748529014, −0.59845240966702638212675825026, −0.53431517164563944697843631683,
0.53431517164563944697843631683, 0.59845240966702638212675825026, 0.948981495105276082949748529014, 1.04098466708431820724940601665, 1.96165698914106258090616703328, 2.12178002997223937500318083580, 2.20953140216932312986264043543, 2.35449120286513803042115668373, 3.08547771765518802723362042798, 3.25989826073943185683030017111, 3.35891943167020866143276455061, 3.61833418753465841126182686572, 4.35984191602282888865110171759, 4.53189417817709244963502839432, 4.54531782365560943770474786961, 4.69665151853057699091628749011, 5.30735160576390507043352340233, 5.48242410780587910399815249741, 5.67566761043521053847052185471, 5.85521341366865022521674848170, 6.06155203519871845593758221394, 6.37719540197043622155254838869, 6.47533439633208189711279216095, 6.92234774100007911147883693007, 7.39000819558517722275022274465
