| L(s) = 1 | + 64·4-s + 4·7-s − 3.83e3·13-s + 1.02e3·16-s + 2.04e4·19-s − 1.84e4·25-s + 256·28-s + 4.00e4·31-s − 4.04e4·37-s − 1.84e5·43-s + 4.70e5·49-s − 2.45e5·52-s − 6.09e5·61-s − 6.55e4·64-s + 2.00e6·67-s + 1.05e6·73-s + 1.31e6·76-s + 8.48e5·79-s − 1.53e4·91-s − 3.62e6·97-s − 1.17e6·100-s − 5.27e6·103-s − 2.48e6·109-s + 4.09e3·112-s − 1.95e6·121-s + 2.56e6·124-s + 127-s + ⋯ |
| L(s) = 1 | + 4-s + 0.0116·7-s − 1.74·13-s + 1/4·16-s + 2.98·19-s − 1.17·25-s + 0.0116·28-s + 1.34·31-s − 0.797·37-s − 2.32·43-s + 3.99·49-s − 1.74·52-s − 2.68·61-s − 1/4·64-s + 6.67·67-s + 2.70·73-s + 2.98·76-s + 1.72·79-s − 0.0203·91-s − 3.96·97-s − 1.17·100-s − 4.82·103-s − 1.91·109-s + 0.00291·112-s − 1.10·121-s + 1.34·124-s + 0.0348·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+3)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(16.87080036\) |
| \(L(\frac12)\) |
\(\approx\) |
\(16.87080036\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - p^{5} T^{2} + p^{10} T^{4} )^{2} \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 736 p^{2} T^{2} + 360097 p^{4} T^{4} - 441343136 p^{6} T^{6} - 331209693824 p^{8} T^{8} - 441343136 p^{18} T^{10} + 360097 p^{28} T^{12} + 736 p^{38} T^{14} + p^{48} T^{16} \) |
| 7 | \( ( 1 - 2 T - 235268 T^{2} + 52 T^{3} + 41511156187 T^{4} + 52 p^{6} T^{5} - 235268 p^{12} T^{6} - 2 p^{18} T^{7} + p^{24} T^{8} )^{2} \) |
| 11 | \( 1 + 1952680 T^{2} + 3083567918158 T^{4} - 10832424911451056000 T^{6} - \)\(21\!\cdots\!77\)\( T^{8} - 10832424911451056000 p^{12} T^{10} + 3083567918158 p^{24} T^{12} + 1952680 p^{36} T^{14} + p^{48} T^{16} \) |
| 13 | \( ( 1 + 1918 T - 3356387 T^{2} - 5022296426 T^{3} + 8438203548124 T^{4} - 5022296426 p^{6} T^{5} - 3356387 p^{12} T^{6} + 1918 p^{18} T^{7} + p^{24} T^{8} )^{2} \) |
| 17 | \( ( 1 - 81233464 T^{2} + 2800570401456999 T^{4} - 81233464 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 19 | \( ( 1 - 5122 T + 92571840 T^{2} - 5122 p^{6} T^{3} + p^{12} T^{4} )^{4} \) |
| 23 | \( 1 + 306194656 T^{2} + 45364969486178110 T^{4} + \)\(13\!\cdots\!04\)\( T^{6} - \)\(19\!\cdots\!21\)\( T^{8} + \)\(13\!\cdots\!04\)\( p^{12} T^{10} + 45364969486178110 p^{24} T^{12} + 306194656 p^{36} T^{14} + p^{48} T^{16} \) |
| 29 | \( 1 - 218801912 T^{2} - 665575825749798599 T^{4} - \)\(12\!\cdots\!32\)\( T^{6} + \)\(36\!\cdots\!64\)\( T^{8} - \)\(12\!\cdots\!32\)\( p^{12} T^{10} - 665575825749798599 p^{24} T^{12} - 218801912 p^{36} T^{14} + p^{48} T^{16} \) |
| 31 | \( ( 1 - 20048 T - 1257936182 T^{2} + 2308504666048 T^{3} + 1610759885481661411 T^{4} + 2308504666048 p^{6} T^{5} - 1257936182 p^{12} T^{6} - 20048 p^{18} T^{7} + p^{24} T^{8} )^{2} \) |
| 37 | \( ( 1 + 10100 T + 5013752475 T^{2} + 10100 p^{6} T^{3} + p^{12} T^{4} )^{4} \) |
| 41 | \( 1 + 2589640888 T^{2} - 11950492095743827346 T^{4} - \)\(68\!\cdots\!36\)\( T^{6} - \)\(31\!\cdots\!25\)\( T^{8} - \)\(68\!\cdots\!36\)\( p^{12} T^{10} - 11950492095743827346 p^{24} T^{12} + 2589640888 p^{36} T^{14} + p^{48} T^{16} \) |
| 43 | \( ( 1 + 92470 T - 316419596 T^{2} - 349130250016940 T^{3} - 15385889618990334485 T^{4} - 349130250016940 p^{6} T^{5} - 316419596 p^{12} T^{6} + 92470 p^{18} T^{7} + p^{24} T^{8} )^{2} \) |
| 47 | \( 1 + 27858666964 T^{2} + \)\(35\!\cdots\!42\)\( T^{4} + \)\(24\!\cdots\!12\)\( p^{2} T^{6} + \)\(15\!\cdots\!79\)\( p^{4} T^{8} + \)\(24\!\cdots\!12\)\( p^{14} T^{10} + \)\(35\!\cdots\!42\)\( p^{24} T^{12} + 27858666964 p^{36} T^{14} + p^{48} T^{16} \) |
| 53 | \( ( 1 - 51925621720 T^{2} + \)\(16\!\cdots\!94\)\( T^{4} - 51925621720 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 59 | \( 1 + 144480622612 T^{2} + \)\(12\!\cdots\!18\)\( T^{4} + \)\(73\!\cdots\!68\)\( T^{6} + \)\(34\!\cdots\!51\)\( T^{8} + \)\(73\!\cdots\!68\)\( p^{12} T^{10} + \)\(12\!\cdots\!18\)\( p^{24} T^{12} + 144480622612 p^{36} T^{14} + p^{48} T^{16} \) |
| 61 | \( ( 1 + 304528 T - 5128503047 T^{2} - 1575915008710448 T^{3} + \)\(21\!\cdots\!16\)\( T^{4} - 1575915008710448 p^{6} T^{5} - 5128503047 p^{12} T^{6} + 304528 p^{18} T^{7} + p^{24} T^{8} )^{2} \) |
| 67 | \( ( 1 - 1004486 T + 576587763532 T^{2} - 252615769683118436 T^{3} + \)\(87\!\cdots\!11\)\( T^{4} - 252615769683118436 p^{6} T^{5} + 576587763532 p^{12} T^{6} - 1004486 p^{18} T^{7} + p^{24} T^{8} )^{2} \) |
| 71 | \( ( 1 - 94765336960 T^{2} + \)\(38\!\cdots\!74\)\( T^{4} - 94765336960 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 73 | \( ( 1 - 262912 T + 52515591 p^{2} T^{2} - 262912 p^{6} T^{3} + p^{12} T^{4} )^{4} \) |
| 79 | \( ( 1 - 424358 T + 116649454564 T^{2} + 179395075645340236 T^{3} - \)\(98\!\cdots\!85\)\( T^{4} + 179395075645340236 p^{6} T^{5} + 116649454564 p^{12} T^{6} - 424358 p^{18} T^{7} + p^{24} T^{8} )^{2} \) |
| 83 | \( 1 + 413441187988 T^{2} + \)\(11\!\cdots\!58\)\( T^{4} - \)\(64\!\cdots\!68\)\( T^{6} - \)\(26\!\cdots\!09\)\( T^{8} - \)\(64\!\cdots\!68\)\( p^{12} T^{10} + \)\(11\!\cdots\!58\)\( p^{24} T^{12} + 413441187988 p^{36} T^{14} + p^{48} T^{16} \) |
| 89 | \( ( 1 - 332250007216 T^{2} + \)\(48\!\cdots\!59\)\( T^{4} - 332250007216 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 97 | \( ( 1 + 1810864 T + 1030883258314 T^{2} + 1054649291167231936 T^{3} + \)\(16\!\cdots\!99\)\( T^{4} + 1054649291167231936 p^{6} T^{5} + 1030883258314 p^{12} T^{6} + 1810864 p^{18} T^{7} + p^{24} T^{8} )^{2} \) |
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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
−4.84613263066478634688569853068, −4.36212220304960894116031636029, −4.24596428752086627774621266271, −4.18627239404344035224339477731, −4.03125914397353907134532791635, −3.74168090690689966546753062254, −3.51229886318853810924615294046, −3.49994927874634060274780474356, −3.45681939014794562178648851978, −3.09359272372174818933377952781, −2.73196538470891222377130174115, −2.71603175419994484376381960300, −2.47406291476136074134517621688, −2.45796904954714194757540929572, −2.27057662456083825419372236583, −2.19924852971607551157292036424, −1.78233249484582640718848110531, −1.45877427017401829768352932781, −1.34918817098490835607613431479, −1.22213026312063924513824729793, −1.17221197598771822830256787170, −0.67716411088750949645448610336, −0.47457504687966383289599796479, −0.39241375109515108283754120049, −0.28564315810198911496395496353,
0.28564315810198911496395496353, 0.39241375109515108283754120049, 0.47457504687966383289599796479, 0.67716411088750949645448610336, 1.17221197598771822830256787170, 1.22213026312063924513824729793, 1.34918817098490835607613431479, 1.45877427017401829768352932781, 1.78233249484582640718848110531, 2.19924852971607551157292036424, 2.27057662456083825419372236583, 2.45796904954714194757540929572, 2.47406291476136074134517621688, 2.71603175419994484376381960300, 2.73196538470891222377130174115, 3.09359272372174818933377952781, 3.45681939014794562178648851978, 3.49994927874634060274780474356, 3.51229886318853810924615294046, 3.74168090690689966546753062254, 4.03125914397353907134532791635, 4.18627239404344035224339477731, 4.24596428752086627774621266271, 4.36212220304960894116031636029, 4.84613263066478634688569853068
Plot not available for L-functions of degree greater than 10.
