L(s) = 1 | + 3·4-s + 5·16-s + 25-s + 20·37-s − 8·43-s − 7·49-s + 3·64-s + 24·67-s + 3·100-s − 28·109-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 60·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 22·169-s − 24·172-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
L(s) = 1 | + 3/2·4-s + 5/4·16-s + 1/5·25-s + 3.28·37-s − 1.21·43-s − 49-s + 3/8·64-s + 2.93·67-s + 3/10·100-s − 2.68·109-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 4.93·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.69·169-s − 1.82·172-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.253375518\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.253375518\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | | \( 1 \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−9.629316970149713957891799882232, −9.266874953140286712491824843895, −8.318071132202098463163702948204, −8.092721354163082201775927126702, −7.57955991739629067136427002537, −6.98769429632661948164430688998, −6.52212535555073673131226209658, −6.21333713448335166339159279635, −5.55697277880198946381540822310, −4.92294771452158522865499193275, −4.18793092031175921202693687448, −3.44255415723091573070298437865, −2.70558977065123798813507920334, −2.22110289569580257150937635230, −1.20596112310386171313122973365,
1.20596112310386171313122973365, 2.22110289569580257150937635230, 2.70558977065123798813507920334, 3.44255415723091573070298437865, 4.18793092031175921202693687448, 4.92294771452158522865499193275, 5.55697277880198946381540822310, 6.21333713448335166339159279635, 6.52212535555073673131226209658, 6.98769429632661948164430688998, 7.57955991739629067136427002537, 8.092721354163082201775927126702, 8.318071132202098463163702948204, 9.266874953140286712491824843895, 9.629316970149713957891799882232