| L(s) = 1 | − 4·5-s + 4·13-s + 4·17-s + 2·25-s + 20·37-s + 8·41-s − 28·53-s + 24·61-s − 16·65-s − 28·73-s + 18·81-s − 16·85-s − 28·97-s − 40·101-s + 36·113-s + 20·121-s + 28·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 1.10·13-s + 0.970·17-s + 2/5·25-s + 3.28·37-s + 1.24·41-s − 3.84·53-s + 3.07·61-s − 1.98·65-s − 3.27·73-s + 2·81-s − 1.73·85-s − 2.84·97-s − 3.98·101-s + 3.38·113-s + 1.81·121-s + 2.50·125-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 + 8/13·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6152318592\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6152318592\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 - p T^{2} )^{2}( 1 + p T^{2} )^{2} \) |
| 7 | $C_2^3$ | \( 1 - 34 T^{4} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 542 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 43 | $C_2^3$ | \( 1 + 2702 T^{4} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 3326 T^{4} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 67 | $C_2^3$ | \( 1 - 2578 T^{4} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2^3$ | \( 1 + 2606 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 114 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−11.01609878409626437252855464725, −10.58256322170514485076474245011, −9.825688468270321541522511683163, −9.757738081556957364521266079889, −9.751647727107471937945948420299, −9.295272979974243628707431541317, −8.798299079799299029076745321181, −8.391230144772941163596719172237, −8.206117683332553297486206698747, −8.000128972104114014046112469308, −7.77689947267413755636464967549, −7.26250094854783193422925026063, −7.24424605309752547319996364290, −6.65081477425201624883865452323, −6.11690165593306375463584261633, −5.97394988416631984998054114805, −5.70724368024664258813509564284, −5.06669545254652433448097839828, −4.44803129710903846605872563753, −4.35629388791130093372305338230, −3.93949304533825941930484225373, −3.42667385456365767720831796605, −3.11052228621699238958233508278, −2.41274550187004575203414873566, −1.21171864887547730782268775920,
1.21171864887547730782268775920, 2.41274550187004575203414873566, 3.11052228621699238958233508278, 3.42667385456365767720831796605, 3.93949304533825941930484225373, 4.35629388791130093372305338230, 4.44803129710903846605872563753, 5.06669545254652433448097839828, 5.70724368024664258813509564284, 5.97394988416631984998054114805, 6.11690165593306375463584261633, 6.65081477425201624883865452323, 7.24424605309752547319996364290, 7.26250094854783193422925026063, 7.77689947267413755636464967549, 8.000128972104114014046112469308, 8.206117683332553297486206698747, 8.391230144772941163596719172237, 8.798299079799299029076745321181, 9.295272979974243628707431541317, 9.751647727107471937945948420299, 9.757738081556957364521266079889, 9.825688468270321541522511683163, 10.58256322170514485076474245011, 11.01609878409626437252855464725
