| L(s) = 1 | − 84·3-s − 1.02e3·4-s + 4.51e4·7-s + 8.30e4·9-s + 8.60e4·12-s − 2.75e5·13-s + 7.86e5·16-s − 1.56e6·19-s − 3.78e6·21-s − 1.83e7·27-s − 4.61e7·28-s − 2.17e7·31-s − 8.50e7·36-s + 7.10e7·37-s + 2.31e7·39-s + 4.70e8·43-s − 6.60e7·48-s + 4.27e8·49-s + 2.81e8·52-s + 1.31e8·57-s − 1.18e9·61-s + 3.74e9·63-s − 5.36e8·64-s + 2.97e8·67-s − 6.53e9·73-s + 1.60e9·76-s + 1.99e8·79-s + ⋯ |
| L(s) = 1 | − 0.345·3-s − 4-s + 2.68·7-s + 1.40·9-s + 0.345·12-s − 0.741·13-s + 3/4·16-s − 0.633·19-s − 0.927·21-s − 1.27·27-s − 2.68·28-s − 0.760·31-s − 1.40·36-s + 1.02·37-s + 0.256·39-s + 3.20·43-s − 0.259·48-s + 1.51·49-s + 0.741·52-s + 0.219·57-s − 1.40·61-s + 3.77·63-s − 1/2·64-s + 0.220·67-s − 3.15·73-s + 0.633·76-s + 0.0647·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(11-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+5)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(9.993807937\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.993807937\) |
| \(L(6)\) |
|
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 | $C_2$ | \( ( 1 + p^{9} T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 + 28 p T - 938 p^{4} T^{2} + 28 p^{11} T^{3} + p^{20} T^{4} \) |
| 5 | | \( 1 \) |
| good | 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
−7.75591532803140889154237088565, −7.53850685072409387554957524605, −7.09348281082224404867007563707, −7.00209119703822747643041474916, −6.37474396967391076727199709168, −6.21098392912219554133165199815, −5.66279532712221641200800959758, −5.55197771532228809131311499033, −5.42351706544423124354609532665, −4.88701230192042489337729329011, −4.53314115514720800505450552414, −4.41309875019210982809589957407, −4.33151112005138653925475220774, −4.29051114786963006138814285027, −3.61586297913753360211912752738, −3.15102041899732652124873586147, −2.98898371408951049657578678796, −2.24814984826177286342936833796, −2.12405714639516977772720686862, −1.74369722639636817065349974674, −1.47934878581305244975120821724, −1.42292055874586782159086811533, −0.62892339417492159160111988290, −0.62522172422253447069246005895, −0.43318599030416082252825453761,
0.43318599030416082252825453761, 0.62522172422253447069246005895, 0.62892339417492159160111988290, 1.42292055874586782159086811533, 1.47934878581305244975120821724, 1.74369722639636817065349974674, 2.12405714639516977772720686862, 2.24814984826177286342936833796, 2.98898371408951049657578678796, 3.15102041899732652124873586147, 3.61586297913753360211912752738, 4.29051114786963006138814285027, 4.33151112005138653925475220774, 4.41309875019210982809589957407, 4.53314115514720800505450552414, 4.88701230192042489337729329011, 5.42351706544423124354609532665, 5.55197771532228809131311499033, 5.66279532712221641200800959758, 6.21098392912219554133165199815, 6.37474396967391076727199709168, 7.00209119703822747643041474916, 7.09348281082224404867007563707, 7.53850685072409387554957524605, 7.75591532803140889154237088565
