| L(s) = 1 | − 2·9-s − 8·11-s − 8·19-s − 10·25-s + 16·29-s + 8·31-s − 4·41-s + 16·49-s + 20·59-s − 36·71-s − 32·79-s + 3·81-s − 32·89-s + 16·99-s + 20·101-s + 48·109-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 24·169-s + ⋯ |
| L(s) = 1 | − 2/3·9-s − 2.41·11-s − 1.83·19-s − 2·25-s + 2.97·29-s + 1.43·31-s − 0.624·41-s + 16/7·49-s + 2.60·59-s − 4.27·71-s − 3.60·79-s + 1/3·81-s − 3.39·89-s + 1.60·99-s + 1.99·101-s + 4.59·109-s − 0.363·121-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 + 1.84·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1705144675\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1705144675\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 142 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 13 | $D_4\times C_2$ | \( 1 - 24 T^{2} + 302 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 31 | $C_4$ | \( ( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 2 T + 38 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 80 T^{2} + 4798 T^{4} + 80 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 2982 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 4822 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 10 T + 98 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 102 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 128 T^{2} + 8574 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 18 T + 218 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 8638 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 16 T + 142 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 220 T^{2} + 22998 T^{4} - 220 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 16 T + 222 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 248 T^{2} + 29694 T^{4} - 248 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.02244475709297635281973488973, −5.70134575522069334691078882688, −5.59072527763094925640632060388, −5.38313141692913691597224538846, −5.17843912155961614641938952091, −5.09300375656129953604384451321, −4.70766003212774932856010075898, −4.42181694466282766178358907042, −4.37241611052468460928019853961, −4.27089181925853376177795912031, −4.10610048722845622415751108265, −3.79966083414469113715176292498, −3.45176441543282188055113069834, −3.03120219789291241890020678953, −3.01425772458401786756405067313, −2.80217576298623238814174035331, −2.67782010055908602708073282445, −2.41172681570718965548775681708, −2.15056542184496250130093119113, −1.98247643508177530252271915590, −1.68185555287550977864723945373, −1.10301304985688485936536160564, −1.04622488441678741255469473672, −0.47714848154106191539011068639, −0.082846234740168725565072584165,
0.082846234740168725565072584165, 0.47714848154106191539011068639, 1.04622488441678741255469473672, 1.10301304985688485936536160564, 1.68185555287550977864723945373, 1.98247643508177530252271915590, 2.15056542184496250130093119113, 2.41172681570718965548775681708, 2.67782010055908602708073282445, 2.80217576298623238814174035331, 3.01425772458401786756405067313, 3.03120219789291241890020678953, 3.45176441543282188055113069834, 3.79966083414469113715176292498, 4.10610048722845622415751108265, 4.27089181925853376177795912031, 4.37241611052468460928019853961, 4.42181694466282766178358907042, 4.70766003212774932856010075898, 5.09300375656129953604384451321, 5.17843912155961614641938952091, 5.38313141692913691597224538846, 5.59072527763094925640632060388, 5.70134575522069334691078882688, 6.02244475709297635281973488973
