| L(s) = 1 | + 12·3-s − 44·7-s + 72·9-s − 296·11-s − 316·13-s + 220·17-s − 528·21-s − 652·23-s + 1.18e3·27-s − 2.76e3·31-s − 3.55e3·33-s + 2.05e3·37-s − 3.79e3·39-s − 4.31e3·41-s + 1.72e3·43-s + 5.25e3·47-s + 968·49-s + 2.64e3·51-s − 6.92e3·53-s − 1.21e4·61-s − 3.16e3·63-s + 6.01e3·67-s − 7.82e3·69-s + 2.23e3·71-s − 1.78e4·73-s + 1.30e4·77-s + 4.05e3·81-s + ⋯ |
| L(s) = 1 | + 4/3·3-s − 0.897·7-s + 8/9·9-s − 2.44·11-s − 1.86·13-s + 0.761·17-s − 1.19·21-s − 1.23·23-s + 1.62·27-s − 2.87·31-s − 3.26·33-s + 1.49·37-s − 2.49·39-s − 2.56·41-s + 0.932·43-s + 2.37·47-s + 0.403·49-s + 1.01·51-s − 2.46·53-s − 3.25·61-s − 0.798·63-s + 1.33·67-s − 1.64·69-s + 0.442·71-s − 3.34·73-s + 2.19·77-s + 0.617·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.456635909\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.456635909\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( ( 1 - 2 p T + 2 p^{2} T^{2} - 2 p^{5} T^{3} + p^{8} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 + 44 T + 968 T^{2} - 19844 p T^{3} - 223634 p^{2} T^{4} - 19844 p^{5} T^{5} + 968 p^{8} T^{6} + 44 p^{12} T^{7} + p^{16} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 148 T + 23158 T^{2} + 148 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 316 T + 49928 T^{2} + 856676 p T^{3} + 14434846 p^{2} T^{4} + 856676 p^{5} T^{5} + 49928 p^{8} T^{6} + 316 p^{12} T^{7} + p^{16} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 220 T + 24200 T^{2} + 710380 T^{3} - 7504709618 T^{4} + 710380 p^{4} T^{5} + 24200 p^{8} T^{6} - 220 p^{12} T^{7} + p^{16} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 427396 T^{2} + 77993999686 T^{4} - 427396 p^{8} T^{6} + p^{16} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 652 T + 212552 T^{2} + 183067908 T^{3} + 157672813838 T^{4} + 183067908 p^{4} T^{5} + 212552 p^{8} T^{6} + 652 p^{12} T^{7} + p^{16} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 851332 T^{2} + 370965009478 T^{4} - 851332 p^{8} T^{6} + p^{16} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 1380 T + 2033142 T^{2} + 1380 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 2052 T + 2105352 T^{2} - 1486261548 T^{3} - 272964536242 T^{4} - 1486261548 p^{4} T^{5} + 2105352 p^{8} T^{6} - 2052 p^{12} T^{7} + p^{16} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 2156 T + 3055206 T^{2} + 2156 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 1724 T + 1486088 T^{2} - 1694904052 T^{3} - 3999465653426 T^{4} - 1694904052 p^{4} T^{5} + 1486088 p^{8} T^{6} - 1724 p^{12} T^{7} + p^{16} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 5252 T + 13791752 T^{2} - 34933252588 T^{3} + 85343693886478 T^{4} - 34933252588 p^{4} T^{5} + 13791752 p^{8} T^{6} - 5252 p^{12} T^{7} + p^{16} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 6924 T + 23970888 T^{2} + 87091477572 T^{3} + 294447643869134 T^{4} + 87091477572 p^{4} T^{5} + 23970888 p^{8} T^{6} + 6924 p^{12} T^{7} + p^{16} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 16072452 T^{2} + 338035836658118 T^{4} - 16072452 p^{8} T^{6} + p^{16} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 6060 T + 23462982 T^{2} + 6060 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 6012 T + 18072072 T^{2} - 131433977268 T^{3} + 952961855450318 T^{4} - 131433977268 p^{4} T^{5} + 18072072 p^{8} T^{6} - 6012 p^{12} T^{7} + p^{16} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 1116 T + 35254326 T^{2} - 1116 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 244 p T + 29768 p^{2} T^{2} + 15649149596 p T^{3} + 6949639479021838 T^{4} + 15649149596 p^{5} T^{5} + 29768 p^{10} T^{6} + 244 p^{13} T^{7} + p^{16} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 60033796 T^{2} + 2261038014449926 T^{4} - 60033796 p^{8} T^{6} + p^{16} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 16332 T + 133367112 T^{2} - 713383376868 T^{3} + 3801627135818318 T^{4} - 713383376868 p^{4} T^{5} + 133367112 p^{8} T^{6} - 16332 p^{12} T^{7} + p^{16} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 156089476 T^{2} + 13455945186102406 T^{4} - 156089476 p^{8} T^{6} + p^{16} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 11812 T + 69761672 T^{2} + 1162925642988 T^{3} + 19287468448798478 T^{4} + 1162925642988 p^{4} T^{5} + 69761672 p^{8} T^{6} + 11812 p^{12} T^{7} + p^{16} T^{8} \) |
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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
−8.322277607744480621689023344041, −8.075542966475353554406489967132, −7.76027080348157763357212348764, −7.68013168951677639421964802223, −7.36112016982547799977491814140, −7.22267038856715602964640693302, −7.13931694838132321666124239493, −6.29369626084751354354739535337, −6.21169657175480113294324945555, −5.98144964697240152300765097885, −5.53475547333505282182806963261, −5.20919109537640567259281003386, −4.87126972656200099504434797386, −4.82249340129072188896161908269, −4.31388281171623696253127698246, −3.88065279264343908263429237676, −3.44735659788003820365513379003, −3.22289677007674935477358898514, −2.80370490822258761139364600802, −2.65640934851210244953993566660, −2.33920060926883574627003423580, −1.82681275201194259259019278560, −1.53516113652815665322984057231, −0.46187094050127873801729262286, −0.27973126508083906074520844002,
0.27973126508083906074520844002, 0.46187094050127873801729262286, 1.53516113652815665322984057231, 1.82681275201194259259019278560, 2.33920060926883574627003423580, 2.65640934851210244953993566660, 2.80370490822258761139364600802, 3.22289677007674935477358898514, 3.44735659788003820365513379003, 3.88065279264343908263429237676, 4.31388281171623696253127698246, 4.82249340129072188896161908269, 4.87126972656200099504434797386, 5.20919109537640567259281003386, 5.53475547333505282182806963261, 5.98144964697240152300765097885, 6.21169657175480113294324945555, 6.29369626084751354354739535337, 7.13931694838132321666124239493, 7.22267038856715602964640693302, 7.36112016982547799977491814140, 7.68013168951677639421964802223, 7.76027080348157763357212348764, 8.075542966475353554406489967132, 8.322277607744480621689023344041
