| L(s) = 1 | − 2·4-s + 8·11-s + 3·16-s − 8·19-s + 8·29-s + 16·41-s − 16·44-s − 40·59-s − 8·61-s − 4·64-s + 16·71-s + 16·76-s + 16·79-s + 14·81-s + 8·101-s + 24·109-s − 16·116-s + 12·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 32·164-s + ⋯ |
| L(s) = 1 | − 4-s + 2.41·11-s + 3/4·16-s − 1.83·19-s + 1.48·29-s + 2.49·41-s − 2.41·44-s − 5.20·59-s − 1.02·61-s − 1/2·64-s + 1.89·71-s + 1.83·76-s + 1.80·79-s + 14/9·81-s + 0.796·101-s + 2.29·109-s − 1.48·116-s + 1.09·121-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 − 2.49·164-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.016166958\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.016166958\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 11 | $D_{4}$ | \( ( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 28 T^{2} + 406 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 32 T^{2} + 706 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 4 T + 40 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 1030 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 3670 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 8 T + 48 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 84 T^{2} + 4310 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 2854 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 5366 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 20 T + 216 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 172 T^{2} + 14326 T^{4} - 172 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 160 T^{2} + 16546 T^{4} - 160 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 8 T + 166 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 288 T^{2} + 34226 T^{4} - 288 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 64 T^{2} + 9474 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.39399130804145172025830358645, −6.09143497799079737905451503493, −6.01108546143869933971750318286, −5.98611343077453752250912950188, −5.56335605917819474829950019010, −5.10599023679728606407900167558, −5.01702092665671347202660439597, −4.89893057642734328137393798047, −4.52926207254757590896453942987, −4.38935363964853185591959538580, −4.35300909641337723094536001973, −4.08159495995994266439823254636, −3.79276227776567919149712200088, −3.60473933700510773234109545827, −3.44368444177730901346324378500, −3.12584128703692414453406016990, −2.77465960323229942562283565960, −2.68033962629407582682350628222, −2.17954991242100634017009984744, −1.94693181847744733460579298353, −1.77503131842460740954774796660, −1.20516095868115670357061344905, −1.14694028055922218507332079025, −0.77166113878200625449024436728, −0.26729954105367526078283590755,
0.26729954105367526078283590755, 0.77166113878200625449024436728, 1.14694028055922218507332079025, 1.20516095868115670357061344905, 1.77503131842460740954774796660, 1.94693181847744733460579298353, 2.17954991242100634017009984744, 2.68033962629407582682350628222, 2.77465960323229942562283565960, 3.12584128703692414453406016990, 3.44368444177730901346324378500, 3.60473933700510773234109545827, 3.79276227776567919149712200088, 4.08159495995994266439823254636, 4.35300909641337723094536001973, 4.38935363964853185591959538580, 4.52926207254757590896453942987, 4.89893057642734328137393798047, 5.01702092665671347202660439597, 5.10599023679728606407900167558, 5.56335605917819474829950019010, 5.98611343077453752250912950188, 6.01108546143869933971750318286, 6.09143497799079737905451503493, 6.39399130804145172025830358645
