| L(s) = 1 | + 3·3-s + 10·7-s + 6·9-s − 9·13-s + 8·19-s + 30·21-s − 5·25-s + 9·27-s − 27·39-s − 13·43-s + 61·49-s + 24·57-s − 61-s + 60·63-s + 21·67-s − 17·73-s − 15·75-s + 9·79-s + 9·81-s − 90·91-s − 24·97-s − 36·109-s − 54·117-s + 22·121-s + 127-s − 39·129-s + 131-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 3.77·7-s + 2·9-s − 2.49·13-s + 1.83·19-s + 6.54·21-s − 25-s + 1.73·27-s − 4.32·39-s − 1.98·43-s + 61/7·49-s + 3.17·57-s − 0.128·61-s + 7.55·63-s + 2.56·67-s − 1.98·73-s − 1.73·75-s + 1.01·79-s + 81-s − 9.43·91-s − 2.43·97-s − 3.44·109-s − 4.99·117-s + 2·121-s + 0.0887·127-s − 3.43·129-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
| \(L(1)\) |
\(\approx\) |
\(6.254104341\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.254104341\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−9.945658478946246537714761087508, −9.928612178167159912809993714179, −9.469820562591393852898259193574, −8.957020556546414183889186123776, −8.333419746413785022011167173882, −8.216998760458442233424620998517, −7.88813762640433061363143145341, −7.56938337420508142754658130787, −7.14651385057017684132040019813, −6.98042654099905501436006492195, −5.54965912198979869561048570285, −5.36620250397231844451617001840, −4.80394077382316967124084758876, −4.68743780324641005896945955857, −4.09801939102367821626202153730, −3.45352670381144064493442280730, −2.56707010043686208432243447665, −2.30585572546382674002330257810, −1.70088139618554623603763509828, −1.28526050084237823454170558749,
1.28526050084237823454170558749, 1.70088139618554623603763509828, 2.30585572546382674002330257810, 2.56707010043686208432243447665, 3.45352670381144064493442280730, 4.09801939102367821626202153730, 4.68743780324641005896945955857, 4.80394077382316967124084758876, 5.36620250397231844451617001840, 5.54965912198979869561048570285, 6.98042654099905501436006492195, 7.14651385057017684132040019813, 7.56938337420508142754658130787, 7.88813762640433061363143145341, 8.216998760458442233424620998517, 8.333419746413785022011167173882, 8.957020556546414183889186123776, 9.469820562591393852898259193574, 9.928612178167159912809993714179, 9.945658478946246537714761087508
