L(s) = 1 | − 4·4-s + 12·16-s − 24·19-s + 4·25-s + 12·31-s − 16·37-s + 14·49-s − 32·64-s + 96·76-s − 16·100-s − 60·103-s + 20·109-s − 20·121-s − 48·124-s + 127-s + 131-s + 137-s + 139-s + 64·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + ⋯ |
L(s) = 1 | − 2·4-s + 3·16-s − 5.50·19-s + 4/5·25-s + 2.15·31-s − 2.63·37-s + 2·49-s − 4·64-s + 11.0·76-s − 8/5·100-s − 5.91·103-s + 1.91·109-s − 1.81·121-s − 4.31·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5.26·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2416665423\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2416665423\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
good | 5 | $C_2^3$ | \( 1 - 4 T^{2} - 9 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 + 20 T^{2} + 279 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 12 T + 67 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 4 T^{2} - 513 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^3$ | \( 1 + 68 T^{2} + 2943 T^{4} + 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 71 | $C_2^3$ | \( 1 - 100 T^{2} + 4959 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 89 | $C_2^3$ | \( 1 - 172 T^{2} + 21663 T^{4} - 172 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
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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
−7.30089097528344957595514972625, −7.21063456998182166636219641828, −6.97958408678587891428006767534, −6.64517430272311836916293408539, −6.40405050592707690154789239509, −6.35870853354464834632140656658, −6.14069549652339257499453065994, −5.61871334706560740919154290932, −5.52079190730766347391884924459, −5.39117058131864075250081933364, −4.87524905566930969665458768793, −4.60809638532206153265447214400, −4.59381120730057052136298734104, −4.36632200498403689343021242560, −3.98161772102204183432411001079, −3.85315419378700559601764685209, −3.75661715970167869441324398042, −3.22130107001465917994455087160, −2.78887369159828521315549200402, −2.42256607588462304624075116247, −2.39490201184095677325762618730, −1.63453292459709642361762182734, −1.54978917119159077659978053998, −0.75695413319349394326302322762, −0.18218392539925573275489855716,
0.18218392539925573275489855716, 0.75695413319349394326302322762, 1.54978917119159077659978053998, 1.63453292459709642361762182734, 2.39490201184095677325762618730, 2.42256607588462304624075116247, 2.78887369159828521315549200402, 3.22130107001465917994455087160, 3.75661715970167869441324398042, 3.85315419378700559601764685209, 3.98161772102204183432411001079, 4.36632200498403689343021242560, 4.59381120730057052136298734104, 4.60809638532206153265447214400, 4.87524905566930969665458768793, 5.39117058131864075250081933364, 5.52079190730766347391884924459, 5.61871334706560740919154290932, 6.14069549652339257499453065994, 6.35870853354464834632140656658, 6.40405050592707690154789239509, 6.64517430272311836916293408539, 6.97958408678587891428006767534, 7.21063456998182166636219641828, 7.30089097528344957595514972625