| L(s) = 1 | + 32·4-s − 778·7-s − 2.83e3·13-s − 1.22e4·19-s − 2.94e4·25-s − 2.48e4·28-s + 2.26e4·31-s + 1.88e5·37-s − 2.90e5·43-s + 3.86e5·49-s − 9.05e4·52-s + 7.00e5·61-s − 3.27e4·64-s − 2.40e5·67-s + 7.00e5·73-s − 3.92e5·76-s + 5.04e5·79-s + 2.20e6·91-s + 2.59e6·97-s − 9.42e5·100-s + 3.14e6·103-s − 1.94e6·109-s + 7.78e5·121-s + 7.25e5·124-s + 127-s + 131-s + 9.54e6·133-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 2.26·7-s − 1.28·13-s − 1.78·19-s − 1.88·25-s − 1.13·28-s + 0.761·31-s + 3.72·37-s − 3.65·43-s + 3.28·49-s − 0.644·52-s + 3.08·61-s − 1/8·64-s − 0.800·67-s + 1.80·73-s − 0.894·76-s + 1.02·79-s + 2.92·91-s + 2.84·97-s − 0.942·100-s + 2.87·103-s − 1.50·109-s + 0.439·121-s + 0.380·124-s + 4.05·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+3)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(3.168032602\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.168032602\) |
| \(L(4)\) |
|
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 | $C_2^2$ | \( 1 - p^{5} T^{2} + p^{10} T^{4} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^3$ | \( 1 + 1178 p^{2} T^{2} + 997059 p^{4} T^{4} + 1178 p^{14} T^{6} + p^{24} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 + 389 T + 33672 T^{2} + 389 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - 778678 T^{2} - 2532088949037 T^{4} - 778678 p^{12} T^{6} + p^{24} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 1415 T - 2824584 T^{2} + 1415 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 42670586 T^{2} + p^{12} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 3067 T + p^{6} T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 - 142126630 T^{2} - 1714645476863421 T^{4} - 142126630 p^{12} T^{6} + p^{24} T^{8} \) |
| 29 | $C_2^3$ | \( 1 + 1021981970 T^{2} + 690632363799611859 T^{4} + 1021981970 p^{12} T^{6} + p^{24} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 - 11338 T - 758953437 T^{2} - 11338 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 47135 T + p^{6} T^{2} )^{4} \) |
| 41 | $C_2^3$ | \( 1 + 9461839682 T^{2} + 66962919867503675043 T^{4} + 9461839682 p^{12} T^{6} + p^{24} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 + 145118 T + 14737870875 T^{2} + 145118 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 10908185542 T^{2} + 2797028709749255523 T^{4} - 10908185542 p^{12} T^{6} + p^{24} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + 26326193614 T^{2} + p^{12} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 46491888310 T^{2} + \)\(38\!\cdots\!19\)\( T^{4} - 46491888310 p^{12} T^{6} + p^{24} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 350305 T + 71193218664 T^{2} - 350305 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 120341 T - 75976425888 T^{2} + 120341 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 143908067234 T^{2} + p^{12} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 175151 T + p^{6} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 252259 T - 179452852440 T^{2} - 252259 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 538899522770 T^{2} + \)\(18\!\cdots\!39\)\( T^{4} + 538899522770 p^{12} T^{6} + p^{24} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 369908068250 T^{2} + p^{12} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 1297105 T + 849509376096 T^{2} - 1297105 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
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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
−8.269889735291045762230086643138, −7.87306648449285120418418752146, −7.76536008061516214933571673696, −7.45311202219600358563901303880, −7.04708889338972959874146853795, −6.64968907469612463304456867717, −6.59109296226721769883920424313, −6.35889275018337499515132976036, −6.19352439112723331060588994263, −5.78155084453566623193421527522, −5.42230522924932487145231378923, −5.17834425596388211360321444792, −4.54805280720455047325942465188, −4.36704602584447569112308818743, −4.17198463795152821356581411196, −3.60018349874754483592120738510, −3.28166674042004454450124813041, −3.17640183575700694616864634103, −2.46044921420821258752194530509, −2.42377009916850530940366122988, −1.98862649485580996682960395134, −1.77142665621155530444265273022, −0.69574310441265592074250664980, −0.48689087649009089066923298043, −0.46420448697845816662695662212,
0.46420448697845816662695662212, 0.48689087649009089066923298043, 0.69574310441265592074250664980, 1.77142665621155530444265273022, 1.98862649485580996682960395134, 2.42377009916850530940366122988, 2.46044921420821258752194530509, 3.17640183575700694616864634103, 3.28166674042004454450124813041, 3.60018349874754483592120738510, 4.17198463795152821356581411196, 4.36704602584447569112308818743, 4.54805280720455047325942465188, 5.17834425596388211360321444792, 5.42230522924932487145231378923, 5.78155084453566623193421527522, 6.19352439112723331060588994263, 6.35889275018337499515132976036, 6.59109296226721769883920424313, 6.64968907469612463304456867717, 7.04708889338972959874146853795, 7.45311202219600358563901303880, 7.76536008061516214933571673696, 7.87306648449285120418418752146, 8.269889735291045762230086643138
