| L(s) = 1 | − 16·23-s + 8·25-s + 8·37-s − 16·47-s + 4·49-s + 32·59-s − 8·61-s − 16·71-s + 16·73-s − 32·83-s + 32·97-s + 32·109-s − 28·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 36·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 3.33·23-s + 8/5·25-s + 1.31·37-s − 2.33·47-s + 4/7·49-s + 4.16·59-s − 1.02·61-s − 1.89·71-s + 1.87·73-s − 3.51·83-s + 3.24·97-s + 3.06·109-s − 2.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2.76·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.338880175\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.338880175\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 34 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $C_4\times C_2$ | \( 1 - 4 T^{2} - 26 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 32 T^{2} + 706 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 104 T^{2} + 4354 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 36 T^{2} + 1094 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 128 T^{2} + 7330 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 3094 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 104 T^{2} + 7522 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 4 T + 94 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 8 T + 86 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 79 | $C_4\times C_2$ | \( 1 - 164 T^{2} + 18054 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 16 T + 222 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 192 T^{2} + 20450 T^{4} - 192 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 16 T + 226 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.05037810678730858280832715059, −6.78670974354203424011367222553, −6.52331597852173763444731355765, −6.25352706325680377581448148604, −6.24315695442239924398328202053, −5.79265631808348482552702571428, −5.71027475907301537924616768440, −5.64686861246427816297408382419, −5.16088655871064618625386181667, −4.88378037441442421833968986480, −4.65549955431119999458359423907, −4.62728601826417642329631424696, −4.25328063149764597039923097418, −3.78676472210939405701254327598, −3.76694970362761823829882908992, −3.70263642532890821053294044803, −3.15419101608530799103926814465, −2.88003846698935813681377839387, −2.62112419134848905602478919957, −2.24933474513764952603908938743, −2.07713242662170756287935472919, −1.74409371386106661551645856375, −1.24760914673966060729512368004, −0.914404346678317626253285303243, −0.27176694672128188998320916940,
0.27176694672128188998320916940, 0.914404346678317626253285303243, 1.24760914673966060729512368004, 1.74409371386106661551645856375, 2.07713242662170756287935472919, 2.24933474513764952603908938743, 2.62112419134848905602478919957, 2.88003846698935813681377839387, 3.15419101608530799103926814465, 3.70263642532890821053294044803, 3.76694970362761823829882908992, 3.78676472210939405701254327598, 4.25328063149764597039923097418, 4.62728601826417642329631424696, 4.65549955431119999458359423907, 4.88378037441442421833968986480, 5.16088655871064618625386181667, 5.64686861246427816297408382419, 5.71027475907301537924616768440, 5.79265631808348482552702571428, 6.24315695442239924398328202053, 6.25352706325680377581448148604, 6.52331597852173763444731355765, 6.78670974354203424011367222553, 7.05037810678730858280832715059
