| L(s) = 1 | + 4-s + 14·13-s + 6·17-s − 12·19-s + 12·23-s − 2·25-s − 12·29-s + 18·37-s + 36·41-s − 4·43-s − 5·49-s + 14·52-s + 12·53-s − 36·59-s − 2·61-s − 64-s + 6·68-s − 12·76-s + 4·79-s + 18·89-s + 12·92-s + 42·97-s − 2·100-s + 18·101-s − 16·103-s + 18·107-s − 12·116-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 3.88·13-s + 1.45·17-s − 2.75·19-s + 2.50·23-s − 2/5·25-s − 2.22·29-s + 2.95·37-s + 5.62·41-s − 0.609·43-s − 5/7·49-s + 1.94·52-s + 1.64·53-s − 4.68·59-s − 0.256·61-s − 1/8·64-s + 0.727·68-s − 1.37·76-s + 0.450·79-s + 1.90·89-s + 1.25·92-s + 4.26·97-s − 1/5·100-s + 1.79·101-s − 1.57·103-s + 1.74·107-s − 1.11·116-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.129061948\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.129061948\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| good | 7 | $C_2^3$ | \( 1 + 5 T^{2} - 24 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 + 13 T^{2} + 48 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 6 T + 20 T^{2} + 108 T^{3} - 645 T^{4} + 108 p T^{5} + 20 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 12 T + 89 T^{2} + 492 T^{3} + 2232 T^{4} + 492 p T^{5} + 89 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 12 T + 74 T^{2} - 288 T^{3} + 1059 T^{4} - 288 p T^{5} + 74 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 12 T + 62 T^{2} + 288 T^{3} + 1707 T^{4} + 288 p T^{5} + 62 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 4314 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 18 T + 173 T^{2} - 1170 T^{3} + 6852 T^{4} - 1170 p T^{5} + 173 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - 18 T + 149 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 - 6 T + 103 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 18 T + 167 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 + 2 T - 92 T^{2} - 52 T^{3} + 5251 T^{4} - 52 p T^{5} - 92 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^3$ | \( 1 + 106 T^{2} + 6195 T^{4} + 106 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 11802 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 2 T + 132 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 260 T^{2} + 29706 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 18 T + 169 T^{2} - 1098 T^{3} + 5412 T^{4} - 1098 p T^{5} + 169 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 42 T + 920 T^{2} - 13944 T^{3} + 157851 T^{4} - 13944 p T^{5} + 920 p^{2} T^{6} - 42 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.14814132730202378956134266091, −6.46236638507043188175731955225, −6.40743304525935326240242282382, −6.27090393275690705864208186506, −6.04178983512528816440843622668, −5.96516665487585841304358656757, −5.94745035776600725950021714712, −5.69647880159147388958192433903, −5.25462217904939245576280213232, −4.72971182057948589542339682706, −4.67754128644579043563812272672, −4.51984528597314185456564094023, −4.26874607948717822400617284335, −3.80336693919473225590623695387, −3.59218600393237613177269188992, −3.54147265847013944583948523597, −3.44714115541783750458615483878, −2.86842051306829180456458894813, −2.48951494358276105127215210534, −2.34648637138910443730258356327, −2.16139790660953337069016226905, −1.36979611474280284050893571697, −1.18881710923897909886936943145, −1.18222837800725907747086127371, −0.55822275020062792126481433454,
0.55822275020062792126481433454, 1.18222837800725907747086127371, 1.18881710923897909886936943145, 1.36979611474280284050893571697, 2.16139790660953337069016226905, 2.34648637138910443730258356327, 2.48951494358276105127215210534, 2.86842051306829180456458894813, 3.44714115541783750458615483878, 3.54147265847013944583948523597, 3.59218600393237613177269188992, 3.80336693919473225590623695387, 4.26874607948717822400617284335, 4.51984528597314185456564094023, 4.67754128644579043563812272672, 4.72971182057948589542339682706, 5.25462217904939245576280213232, 5.69647880159147388958192433903, 5.94745035776600725950021714712, 5.96516665487585841304358656757, 6.04178983512528816440843622668, 6.27090393275690705864208186506, 6.40743304525935326240242282382, 6.46236638507043188175731955225, 7.14814132730202378956134266091
