| L(s) = 1 | − 16·11-s + 4·13-s + 16·23-s + 16·37-s − 24·47-s + 22·49-s − 8·59-s + 4·61-s + 24·71-s − 32·73-s − 24·83-s + 36·97-s + 24·107-s + 44·109-s + 120·121-s + 127-s + 131-s + 137-s + 139-s − 64·143-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + ⋯ |
| L(s) = 1 | − 4.82·11-s + 1.10·13-s + 3.33·23-s + 2.63·37-s − 3.50·47-s + 22/7·49-s − 1.04·59-s + 0.512·61-s + 2.84·71-s − 3.74·73-s − 2.63·83-s + 3.65·97-s + 2.32·107-s + 4.21·109-s + 10.9·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5.35·143-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.859125873\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.859125873\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 22 T^{2} + 211 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 8 T + 36 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_{4}$ | \( ( 1 - 2 T - 5 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 56 T^{2} + 1330 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 22 T^{2} + 195 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 8 T + 60 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 8 T^{2} - 894 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 86 T^{2} + 3699 T^{4} - 86 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 8 T + 82 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 8134 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 118 T^{2} + 6979 T^{4} - 118 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 12 T + 128 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 188 T^{2} + 14326 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 4 T + 72 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 2 T + 115 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 230 T^{2} + 22131 T^{4} - 230 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 12 T + 146 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 16 T + 138 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 + 12 T + 200 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{4} \) |
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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
−5.59913315336096070182540384029, −5.36198875534597758902456013450, −5.08409472427899078738274441792, −4.98349856849110734636748541898, −4.92504300056058751873831524075, −4.75951200118820887232474284413, −4.43226791724868203331082988113, −4.34488712561200073247184536593, −4.33059145868896031290073462302, −3.69103479242007134408347405702, −3.53887867411739176625892938354, −3.36792562242174965657467151721, −3.17198592569700266008301311653, −2.93660792828993910499531775638, −2.80100156972463058883701888261, −2.66538923387610898329670939853, −2.62387606325425864375779872549, −2.18739844854904054825629167632, −1.98450775448176746071945182712, −1.64492141058358606638989812973, −1.61379356915973691451341248411, −0.863846918171139973333220011533, −0.73391924644291854916889066924, −0.67920104162489992579883489351, −0.30017526792653881900534100731,
0.30017526792653881900534100731, 0.67920104162489992579883489351, 0.73391924644291854916889066924, 0.863846918171139973333220011533, 1.61379356915973691451341248411, 1.64492141058358606638989812973, 1.98450775448176746071945182712, 2.18739844854904054825629167632, 2.62387606325425864375779872549, 2.66538923387610898329670939853, 2.80100156972463058883701888261, 2.93660792828993910499531775638, 3.17198592569700266008301311653, 3.36792562242174965657467151721, 3.53887867411739176625892938354, 3.69103479242007134408347405702, 4.33059145868896031290073462302, 4.34488712561200073247184536593, 4.43226791724868203331082988113, 4.75951200118820887232474284413, 4.92504300056058751873831524075, 4.98349856849110734636748541898, 5.08409472427899078738274441792, 5.36198875534597758902456013450, 5.59913315336096070182540384029
