| L(s) = 1 | + 4·3-s + 4·9-s − 4·27-s + 24·31-s − 24·37-s + 24·41-s + 20·43-s + 4·49-s − 20·67-s − 24·71-s − 48·79-s − 10·81-s + 12·83-s + 24·89-s + 96·93-s + 60·107-s − 96·111-s + 20·121-s + 96·123-s + 127-s + 80·129-s + 131-s + 137-s + 139-s + 16·147-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 4/3·9-s − 0.769·27-s + 4.31·31-s − 3.94·37-s + 3.74·41-s + 3.04·43-s + 4/7·49-s − 2.44·67-s − 2.84·71-s − 5.40·79-s − 1.11·81-s + 1.31·83-s + 2.54·89-s + 9.95·93-s + 5.80·107-s − 9.11·111-s + 1.81·121-s + 8.65·123-s + 0.0887·127-s + 7.04·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.31·147-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.635254862\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.635254862\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( ( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 4 T^{2} - 6 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 186 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $D_{4}$ | \( ( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 41 | $D_{4}$ | \( ( 1 - 12 T + 106 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 10 T + 84 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 6810 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 1974 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 10 T + 132 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 12 T + 70 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 4230 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 83 | $D_{4}$ | \( ( 1 - 6 T + 172 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 16806 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.78240567349042639481273147221, −6.38153684760235733368471546754, −6.09601753356249326983253645477, −6.09107836170262905874907901318, −6.00179148015105020610291146294, −5.70049298852705456716492347727, −5.62182683913409703939460422760, −4.94660877963680126888018262217, −4.78787332436533268920002738291, −4.64654431076956445565122811099, −4.56130022949123542105605598728, −4.19688731453951213127205559124, −4.01952647958183072133863350664, −3.59926156838631284885897468092, −3.42490921202614560283341593560, −3.22867147104805207472808013420, −2.90645131554751467910927678800, −2.67445270862927091414195453660, −2.64447588045411647119143316734, −2.26450784827701981890912682445, −2.25810805864677933439854880971, −1.60450781011561824536650431566, −1.11715524097379621428928405349, −1.10555075908559938036157228688, −0.37995873768956443100235724700,
0.37995873768956443100235724700, 1.10555075908559938036157228688, 1.11715524097379621428928405349, 1.60450781011561824536650431566, 2.25810805864677933439854880971, 2.26450784827701981890912682445, 2.64447588045411647119143316734, 2.67445270862927091414195453660, 2.90645131554751467910927678800, 3.22867147104805207472808013420, 3.42490921202614560283341593560, 3.59926156838631284885897468092, 4.01952647958183072133863350664, 4.19688731453951213127205559124, 4.56130022949123542105605598728, 4.64654431076956445565122811099, 4.78787332436533268920002738291, 4.94660877963680126888018262217, 5.62182683913409703939460422760, 5.70049298852705456716492347727, 6.00179148015105020610291146294, 6.09107836170262905874907901318, 6.09601753356249326983253645477, 6.38153684760235733368471546754, 6.78240567349042639481273147221
