| L(s) = 1 | − 6·7-s − 6·11-s + 8·17-s − 6·19-s − 2·23-s + 6·25-s − 2·29-s − 12·37-s − 36·41-s + 2·43-s + 8·49-s + 12·53-s + 8·61-s + 42·67-s + 6·71-s + 36·77-s + 24·79-s + 12·89-s + 10·101-s − 68·103-s − 10·107-s − 20·113-s − 48·119-s + 8·121-s + 127-s + 131-s + 36·133-s + ⋯ |
| L(s) = 1 | − 2.26·7-s − 1.80·11-s + 1.94·17-s − 1.37·19-s − 0.417·23-s + 6/5·25-s − 0.371·29-s − 1.97·37-s − 5.62·41-s + 0.304·43-s + 8/7·49-s + 1.64·53-s + 1.02·61-s + 5.13·67-s + 0.712·71-s + 4.10·77-s + 2.70·79-s + 1.27·89-s + 0.995·101-s − 6.70·103-s − 0.966·107-s − 1.88·113-s − 4.40·119-s + 8/11·121-s + 0.0887·127-s + 0.0873·131-s + 3.12·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7768911304\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7768911304\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| good | 5 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} ) \) |
| 7 | $D_4\times C_2$ | \( 1 + 6 T + 4 p T^{2} + 96 T^{3} + 291 T^{4} + 96 p T^{5} + 4 p^{3} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 6 T + 28 T^{2} + 96 T^{3} + 267 T^{4} + 96 p T^{5} + 28 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2$$\times$$C_2^2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $D_4\times C_2$ | \( 1 + 6 T + 44 T^{2} + 192 T^{3} + 891 T^{4} + 192 p T^{5} + 44 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 2 T - 16 T^{2} - 52 T^{3} - 221 T^{4} - 52 p T^{5} - 16 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 2 T - 43 T^{2} - 22 T^{3} + 1252 T^{4} - 22 p T^{5} - 43 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 92 T^{2} + 3846 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + 48 p T^{4} + 12 p^{2} T^{5} + 85 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 36 T + 621 T^{2} + 6804 T^{3} + 51752 T^{4} + 6804 p T^{5} + 621 p^{2} T^{6} + 36 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 2 T - 8 T^{2} + 148 T^{3} - 1877 T^{4} + 148 p T^{5} - 8 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 6810 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 6 T + 103 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 + 54 T^{2} - 565 T^{4} + 54 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 8 T - 47 T^{2} + 88 T^{3} + 4696 T^{4} + 88 p T^{5} - 47 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 42 T + 868 T^{2} - 11760 T^{3} + 113307 T^{4} - 11760 p T^{5} + 868 p^{2} T^{6} - 42 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 6 T + 148 T^{2} - 816 T^{3} + 14307 T^{4} - 816 p T^{5} + 148 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 158 T^{2} + 16131 T^{4} - 158 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 12 T + 182 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 228 T^{2} + 24074 T^{4} - 228 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 12 T + 202 T^{2} - 1848 T^{3} + 20067 T^{4} - 1848 p T^{5} + 202 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^3$ | \( 1 + 158 T^{2} + 15555 T^{4} + 158 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.59600599287474826796207657795, −6.54184588508212039654630994599, −6.10319589174468896248413298343, −6.08173485506847640583162557992, −5.58745827552175255440723903903, −5.36713891312074962798839302427, −5.26895655350008506352733111239, −5.18651696310599852709536338531, −5.07717583428523280393702250212, −4.76767583284642351725403674986, −4.45967987150735224544544887405, −3.75712707449025414375243915440, −3.74139730507250307710627469207, −3.73781223880333018593921508335, −3.54001239452120043864182598388, −3.16873347263288901496358235640, −3.14276534913239257187332781737, −2.62558834582197104254129589859, −2.34667998785262292360818744669, −2.31863657989459619960299699898, −1.96075612473528312216246045724, −1.37604663740979193412196102443, −1.23330870726592085771285865537, −0.36515766398365490007487353995, −0.34156125650097746099352801886,
0.34156125650097746099352801886, 0.36515766398365490007487353995, 1.23330870726592085771285865537, 1.37604663740979193412196102443, 1.96075612473528312216246045724, 2.31863657989459619960299699898, 2.34667998785262292360818744669, 2.62558834582197104254129589859, 3.14276534913239257187332781737, 3.16873347263288901496358235640, 3.54001239452120043864182598388, 3.73781223880333018593921508335, 3.74139730507250307710627469207, 3.75712707449025414375243915440, 4.45967987150735224544544887405, 4.76767583284642351725403674986, 5.07717583428523280393702250212, 5.18651696310599852709536338531, 5.26895655350008506352733111239, 5.36713891312074962798839302427, 5.58745827552175255440723903903, 6.08173485506847640583162557992, 6.10319589174468896248413298343, 6.54184588508212039654630994599, 6.59600599287474826796207657795
