| L(s) = 1 | + 84·3-s + 4.51e4·7-s + 8.30e4·9-s − 2.75e5·13-s − 1.56e6·19-s + 3.78e6·21-s + 2.66e6·25-s + 1.83e7·27-s + 2.17e7·31-s + 7.10e7·37-s − 2.31e7·39-s − 4.70e8·43-s + 4.27e8·49-s − 1.31e8·57-s + 1.18e9·61-s + 3.74e9·63-s − 2.97e8·67-s + 6.53e9·73-s + 2.23e8·75-s − 1.99e8·79-s + 4.77e9·81-s − 1.24e10·91-s + 1.83e9·93-s − 3.91e10·97-s − 1.10e10·103-s + 2.03e10·109-s + 5.96e9·111-s + ⋯ |
| L(s) = 1 | + 0.345·3-s + 2.68·7-s + 1.40·9-s − 0.741·13-s − 0.633·19-s + 0.927·21-s + 0.272·25-s + 1.27·27-s + 0.760·31-s + 1.02·37-s − 0.256·39-s − 3.20·43-s + 1.51·49-s − 0.219·57-s + 1.40·61-s + 3.77·63-s − 0.220·67-s + 3.15·73-s + 0.0943·75-s − 0.0647·79-s + 1.37·81-s − 1.98·91-s + 0.263·93-s − 4.56·97-s − 0.954·103-s + 1.32·109-s + 0.354·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(11-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+5)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(11.73571621\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.73571621\) |
| \(L(6)\) |
|
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 | $C_2^2$ | \( 1 - 28 p T - 938 p^{4} T^{2} - 28 p^{11} T^{3} + p^{20} T^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 533012 p T^{2} + 6693053487126 p^{2} T^{4} - 533012 p^{21} T^{6} + p^{40} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 - 22556 T + 78482346 p T^{2} - 22556 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 4130589740 p T^{2} + \)\(12\!\cdots\!02\)\( T^{4} - 4130589740 p^{21} T^{6} + p^{40} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 137620 T + 223344855798 T^{2} + 137620 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 503008419460 T^{2} - \)\(28\!\cdots\!38\)\( T^{4} - 503008419460 p^{20} T^{6} + p^{40} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 784364 T + 11072641146486 T^{2} + 784364 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 132650537376580 T^{2} + \)\(14\!\cdots\!78\)\( p^{2} T^{4} - 132650537376580 p^{20} T^{6} + p^{40} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 775524833463844 T^{2} + \)\(31\!\cdots\!86\)\( T^{4} - 775524833463844 p^{20} T^{6} + p^{40} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 10892924 T + 345177991175046 T^{2} - 10892924 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 35507084 T + 5319849690001302 T^{2} - 35507084 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 30129232679351620 T^{2} + \)\(47\!\cdots\!42\)\( T^{4} - 30129232679351620 p^{20} T^{6} + p^{40} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 5473124 p T + 50501107105165014 T^{2} + 5473124 p^{11} T^{3} + p^{20} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 185271765343089796 T^{2} + \)\(13\!\cdots\!06\)\( T^{4} - 185271765343089796 p^{20} T^{6} + p^{40} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 486913415004520420 T^{2} + \)\(10\!\cdots\!62\)\( T^{4} - 486913415004520420 p^{20} T^{6} + p^{40} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 1720402455795118180 T^{2} + \)\(12\!\cdots\!62\)\( T^{4} - 1720402455795118180 p^{20} T^{6} + p^{40} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 592019372 T + 1459215526973846838 T^{2} - 592019372 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 148682924 T + 3295467847639477302 T^{2} + 148682924 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 11173485987747195844 T^{2} + \)\(52\!\cdots\!86\)\( T^{4} - 11173485987747195844 p^{20} T^{6} + p^{40} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 3267134500 T + 10958748572669742438 T^{2} - 3267134500 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 99641284 T + 15267587176773126726 T^{2} + 99641284 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 56087025146752445092 T^{2} + \)\(12\!\cdots\!58\)\( T^{4} - 56087025146752445092 p^{20} T^{6} + p^{40} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 97642754408129363140 T^{2} + \)\(41\!\cdots\!62\)\( T^{4} - 97642754408129363140 p^{20} T^{6} + p^{40} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 19588177532 T + \)\(24\!\cdots\!94\)\( T^{2} + 19588177532 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
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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
−7.32665656466046826236131285085, −7.13355117650317986624490376361, −6.72305272954667597238314049383, −6.58101795147130437623267628498, −6.43245027789842345029052175845, −5.89600759491835282651997173530, −5.41734753088396967498813547916, −5.36939854324026970416066476663, −4.79268217449974299983473311373, −4.71803553128641788939446997497, −4.70139759589804764574248632708, −4.55787359187039846350009073756, −3.82187960124992596012363110338, −3.72754170224257722961055606495, −3.56908189253170827078516892934, −2.83007677399952487012573092861, −2.55061966285347703779829270537, −2.54219500353257052941683496170, −1.83467352148474514857011694273, −1.76115279320876141605324021915, −1.57833705181264508389992955616, −1.30267130746465844141741341327, −0.927543187069921424963790514847, −0.58123278671160711818418598925, −0.28036671028042017912498655810,
0.28036671028042017912498655810, 0.58123278671160711818418598925, 0.927543187069921424963790514847, 1.30267130746465844141741341327, 1.57833705181264508389992955616, 1.76115279320876141605324021915, 1.83467352148474514857011694273, 2.54219500353257052941683496170, 2.55061966285347703779829270537, 2.83007677399952487012573092861, 3.56908189253170827078516892934, 3.72754170224257722961055606495, 3.82187960124992596012363110338, 4.55787359187039846350009073756, 4.70139759589804764574248632708, 4.71803553128641788939446997497, 4.79268217449974299983473311373, 5.36939854324026970416066476663, 5.41734753088396967498813547916, 5.89600759491835282651997173530, 6.43245027789842345029052175845, 6.58101795147130437623267628498, 6.72305272954667597238314049383, 7.13355117650317986624490376361, 7.32665656466046826236131285085
