| L(s) = 1 | + 8·11-s − 16·19-s − 4·25-s − 32·41-s − 16·49-s − 32·67-s + 16·73-s + 8·83-s − 8·89-s + 32·97-s − 16·107-s − 72·113-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 16·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 2.41·11-s − 3.67·19-s − 4/5·25-s − 4.99·41-s − 2.28·49-s − 3.90·67-s + 1.87·73-s + 0.878·83-s − 0.847·89-s + 3.24·97-s − 1.54·107-s − 6.77·113-s + 1.81·121-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 − 1.23·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $D_4$ | \( 1 + 4 T^{2} + 6 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $D_{4}$ | \( ( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 + 4 T^{2} + 1254 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 64 T^{2} + 2034 T^{4} + 64 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 16 T + 134 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 + 100 T^{2} + 6918 T^{4} + 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 + 160 T^{2} + 12114 T^{4} + 160 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 20 T^{2} - 2106 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 152 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 4 T + 158 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 4 T + 134 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 16 T + 210 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.01598433056737367616814791716, −5.51930580061132401176983343779, −5.24896804515026262048818415478, −5.22065525294860821838680640370, −5.03984612445584751386675329635, −4.92326270139031172080576139404, −4.62205581899848941529757752759, −4.42763905646509803507437508484, −4.33414101595569925089617830319, −3.99398378029362401794431652834, −3.97429355907882679609985055539, −3.74197667478715081529198557643, −3.73414631014121962182230348483, −3.26944778456835721062646446024, −3.25636404488577100441318554128, −3.09889389179598106124132983286, −2.72157033237434518888690465060, −2.37891667223088474243754825198, −2.16572247107096338296043164658, −2.03925025985158664096289788396, −2.00818264091025983742851579167, −1.37904217099109950865819985758, −1.36877668485232437639124147929, −1.34502652570077379521728212303, −1.10461950996901492739779869541, 0, 0, 0, 0,
1.10461950996901492739779869541, 1.34502652570077379521728212303, 1.36877668485232437639124147929, 1.37904217099109950865819985758, 2.00818264091025983742851579167, 2.03925025985158664096289788396, 2.16572247107096338296043164658, 2.37891667223088474243754825198, 2.72157033237434518888690465060, 3.09889389179598106124132983286, 3.25636404488577100441318554128, 3.26944778456835721062646446024, 3.73414631014121962182230348483, 3.74197667478715081529198557643, 3.97429355907882679609985055539, 3.99398378029362401794431652834, 4.33414101595569925089617830319, 4.42763905646509803507437508484, 4.62205581899848941529757752759, 4.92326270139031172080576139404, 5.03984612445584751386675329635, 5.22065525294860821838680640370, 5.24896804515026262048818415478, 5.51930580061132401176983343779, 6.01598433056737367616814791716
