| L(s) = 1 | + 2·3-s + 2·5-s + 9-s − 2·11-s + 4·15-s + 2·17-s + 6·19-s + 2·23-s + 25-s − 2·27-s + 20·29-s + 2·31-s − 4·33-s + 4·37-s + 12·41-s + 8·43-s + 2·45-s + 12·47-s + 4·51-s + 4·53-s − 4·55-s + 12·57-s + 6·59-s + 16·61-s + 4·67-s + 4·69-s + 44·71-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s + 1/3·9-s − 0.603·11-s + 1.03·15-s + 0.485·17-s + 1.37·19-s + 0.417·23-s + 1/5·25-s − 0.384·27-s + 3.71·29-s + 0.359·31-s − 0.696·33-s + 0.657·37-s + 1.87·41-s + 1.21·43-s + 0.298·45-s + 1.75·47-s + 0.560·51-s + 0.549·53-s − 0.539·55-s + 1.58·57-s + 0.781·59-s + 2.04·61-s + 0.488·67-s + 0.481·69-s + 5.22·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(19.63508575\) |
| \(L(\frac12)\) |
\(\approx\) |
\(19.63508575\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 11 | $D_4\times C_2$ | \( 1 + 2 T - 12 T^{2} - 12 T^{3} + 91 T^{4} - 12 p T^{5} - 12 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 19 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 2 T - 24 T^{2} + 12 T^{3} + 427 T^{4} + 12 p T^{5} - 24 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 6 T + 17 T^{2} + 6 p T^{3} - 36 p T^{4} + 6 p^{2} T^{5} + 17 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 2 T - 36 T^{2} + 12 T^{3} + 979 T^{4} + 12 p T^{5} - 36 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 10 T + 76 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 2 T - p T^{2} + 54 T^{3} + 140 T^{4} + 54 p T^{5} - p^{3} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 4 T - 55 T^{2} + 12 T^{3} + 3080 T^{4} + 12 p T^{5} - 55 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 4 T + 27 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 4 T - 66 T^{2} + 96 T^{3} + 3067 T^{4} + 96 p T^{5} - 66 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 6 T - 28 T^{2} + 324 T^{3} - 1509 T^{4} + 324 p T^{5} - 28 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 4 T - 115 T^{2} + 12 T^{3} + 11600 T^{4} + 12 p T^{5} - 115 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 22 T + 256 T^{2} - 22 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^3$ | \( 1 + 29 T^{2} - 4488 T^{4} + 29 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 6 T - 103 T^{2} + 114 T^{3} + 10236 T^{4} + 114 p T^{5} - 103 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 6 T + 112 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 2 T - 348 T^{3} - 8261 T^{4} - 348 p T^{5} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
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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
−6.25668516130070046824792996135, −5.96712907808048652741072610139, −5.75248944805588621255045944045, −5.65691514907184834118000207434, −5.43682007160980287065159545501, −4.97914168325719814637447816266, −4.91476910805713739448337287771, −4.89333694703011103219181705200, −4.82314435419983571841355968577, −4.05643603107130952931830118010, −4.04036076174617339072124236578, −4.03006139015354495812346538081, −3.71832979872469511553207423671, −3.22277030184530016251691626455, −3.11748252177924021803271974953, −3.05806467347491425629041466476, −2.66696997131315755703926017989, −2.43700220885002287821600739176, −2.22620771641580654064672179646, −2.14146467057521404832607223079, −2.01039179605024059468063670494, −1.08502021258373310026843029183, −0.941404234055516000169391459426, −0.835232358830141746612723653538, −0.76163827925805889287953545926,
0.76163827925805889287953545926, 0.835232358830141746612723653538, 0.941404234055516000169391459426, 1.08502021258373310026843029183, 2.01039179605024059468063670494, 2.14146467057521404832607223079, 2.22620771641580654064672179646, 2.43700220885002287821600739176, 2.66696997131315755703926017989, 3.05806467347491425629041466476, 3.11748252177924021803271974953, 3.22277030184530016251691626455, 3.71832979872469511553207423671, 4.03006139015354495812346538081, 4.04036076174617339072124236578, 4.05643603107130952931830118010, 4.82314435419983571841355968577, 4.89333694703011103219181705200, 4.91476910805713739448337287771, 4.97914168325719814637447816266, 5.43682007160980287065159545501, 5.65691514907184834118000207434, 5.75248944805588621255045944045, 5.96712907808048652741072610139, 6.25668516130070046824792996135
