L(s) = 1 | + 4·19-s + 8·25-s − 8·31-s + 10·49-s − 52·61-s − 20·79-s + 32·109-s − 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 34·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
L(s) = 1 | + 0.917·19-s + 8/5·25-s − 1.43·31-s + 10/7·49-s − 6.65·61-s − 2.25·79-s + 3.06·109-s − 0.727·121-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 + 2.61·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.055140711\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.055140711\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
good | 7 | $C_2^2$ | \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 17 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 64 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 125 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 65 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 164 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 185 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.48763420572749446520304207625, −6.26801554792319757406757778921, −5.92298764961018639483372774224, −5.81392971546344827669767195182, −5.65111146848112257354826223966, −5.54230579613757813545824362147, −5.14083940833835182768076769999, −4.85884780578900376375661737750, −4.82280637084575706822080377461, −4.51518465739885644886076951116, −4.38204568320391849710925709148, −4.08921211569861187159800963719, −3.97788520306310568290775809653, −3.38602517921363753763135758346, −3.34312120071628030093462920203, −3.10262559583268932757745261390, −2.88746944721734086384559864569, −2.87910346371919771687798427180, −2.28117007132935914284038724616, −1.98564635760011390066467398133, −1.76002275331648304430222086535, −1.49420487921238821708445929026, −1.08510187678724062339329611421, −0.832724645568873429735610497587, −0.18121793090740487203286781442,
0.18121793090740487203286781442, 0.832724645568873429735610497587, 1.08510187678724062339329611421, 1.49420487921238821708445929026, 1.76002275331648304430222086535, 1.98564635760011390066467398133, 2.28117007132935914284038724616, 2.87910346371919771687798427180, 2.88746944721734086384559864569, 3.10262559583268932757745261390, 3.34312120071628030093462920203, 3.38602517921363753763135758346, 3.97788520306310568290775809653, 4.08921211569861187159800963719, 4.38204568320391849710925709148, 4.51518465739885644886076951116, 4.82280637084575706822080377461, 4.85884780578900376375661737750, 5.14083940833835182768076769999, 5.54230579613757813545824362147, 5.65111146848112257354826223966, 5.81392971546344827669767195182, 5.92298764961018639483372774224, 6.26801554792319757406757778921, 6.48763420572749446520304207625