| L(s) = 1 | + 2·5-s − 2·9-s − 2·13-s + 3·25-s − 2·29-s − 2·41-s − 4·45-s − 49-s − 2·61-s − 4·65-s + 3·81-s − 2·97-s + 2·101-s + 2·113-s + 4·117-s − 121-s + 6·125-s + ⋯ |
| L(s) = 1 | + 2·5-s − 2·9-s − 2·13-s + 3·25-s − 2·29-s − 2·41-s − 4·45-s − 49-s − 2·61-s − 4·65-s + 3·81-s − 2·97-s + 2·101-s + 2·113-s + 4·117-s − 121-s + 6·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3857062724\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3857062724\) |
| \(L(1)\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 7 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 31 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 47 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 79 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 83 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
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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
−8.928798352936314543034115685053, −8.649405902211235628701397942728, −8.451192857155567513986596139287, −8.101475624121643854016792628220, −7.935011323989461731400087374067, −7.44841345575103734602400242998, −7.24802279991529912015663167216, −7.12433851161083276572159278385, −6.74677388518014076294327595665, −6.33936217767341830227216817971, −6.12897633470547152576738752297, −5.95266405778449499381373053264, −5.77152792467016302946751352864, −5.28586289209185361498722399010, −5.08644532893701287405730579469, −4.92680694316637013227842401028, −4.85547510927867857249367349822, −4.14404139882933801339493024789, −3.75521121491802393346092585118, −3.12918914925627571539520560095, −3.07175822990398496683217257444, −2.77031961470767503872629884662, −2.16205100118105999082960142639, −2.03612772589127396511352414390, −1.57218322916192363344311116267,
1.57218322916192363344311116267, 2.03612772589127396511352414390, 2.16205100118105999082960142639, 2.77031961470767503872629884662, 3.07175822990398496683217257444, 3.12918914925627571539520560095, 3.75521121491802393346092585118, 4.14404139882933801339493024789, 4.85547510927867857249367349822, 4.92680694316637013227842401028, 5.08644532893701287405730579469, 5.28586289209185361498722399010, 5.77152792467016302946751352864, 5.95266405778449499381373053264, 6.12897633470547152576738752297, 6.33936217767341830227216817971, 6.74677388518014076294327595665, 7.12433851161083276572159278385, 7.24802279991529912015663167216, 7.44841345575103734602400242998, 7.935011323989461731400087374067, 8.101475624121643854016792628220, 8.451192857155567513986596139287, 8.649405902211235628701397942728, 8.928798352936314543034115685053
