| L(s) = 1 | − 3·3-s − 2·7-s + 6·9-s + 8·19-s + 6·21-s − 2·25-s − 9·27-s − 18·29-s − 4·43-s − 11·49-s + 18·53-s − 24·57-s − 6·59-s − 16·61-s − 12·63-s − 24·71-s + 22·73-s + 6·75-s + 9·81-s + 54·87-s − 12·89-s + 6·107-s + 24·113-s + 10·121-s + 127-s + 12·129-s + 131-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.755·7-s + 2·9-s + 1.83·19-s + 1.30·21-s − 2/5·25-s − 1.73·27-s − 3.34·29-s − 0.609·43-s − 1.57·49-s + 2.47·53-s − 3.17·57-s − 0.781·59-s − 2.04·61-s − 1.51·63-s − 2.84·71-s + 2.57·73-s + 0.692·75-s + 81-s + 5.78·87-s − 1.27·89-s + 0.580·107-s + 2.25·113-s + 0.909·121-s + 0.0887·127-s + 1.05·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5053710380\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5053710380\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 19 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 59 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.50687907800007814205506569116, −9.778709980490372354394574336033, −9.594327308765145281382590752132, −9.358588007630277367990214762527, −8.758646213835674476160718874746, −8.050569957155588787338996736996, −7.51993362264571322451593200575, −7.16953367436832341250413445637, −7.04780362551873855628173892357, −6.16856297446901935037821256248, −6.01787168329570719286958867729, −5.51609030431458289553462675499, −5.27897725139969926680995703297, −4.64152189035303064600392259549, −4.12571298516247816773420089958, −3.43549569003335863127305592153, −3.17224876554612289124470968074, −2.00054907663525184032548186890, −1.41257393692893365059355368389, −0.39420738451292526617359265397,
0.39420738451292526617359265397, 1.41257393692893365059355368389, 2.00054907663525184032548186890, 3.17224876554612289124470968074, 3.43549569003335863127305592153, 4.12571298516247816773420089958, 4.64152189035303064600392259549, 5.27897725139969926680995703297, 5.51609030431458289553462675499, 6.01787168329570719286958867729, 6.16856297446901935037821256248, 7.04780362551873855628173892357, 7.16953367436832341250413445637, 7.51993362264571322451593200575, 8.050569957155588787338996736996, 8.758646213835674476160718874746, 9.358588007630277367990214762527, 9.594327308765145281382590752132, 9.778709980490372354394574336033, 10.50687907800007814205506569116
