L(s) = 1 | + 4·7-s + 4·13-s + 8·19-s − 2·25-s + 4·31-s − 12·37-s − 8·43-s + 2·49-s − 12·61-s + 12·73-s + 4·79-s + 16·91-s − 20·97-s + 12·103-s − 12·109-s + 10·121-s + 127-s + 131-s + 32·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + ⋯ |
L(s) = 1 | + 1.51·7-s + 1.10·13-s + 1.83·19-s − 2/5·25-s + 0.718·31-s − 1.97·37-s − 1.21·43-s + 2/7·49-s − 1.53·61-s + 1.40·73-s + 0.450·79-s + 1.67·91-s − 2.03·97-s + 1.18·103-s − 1.14·109-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 2.77·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2/13·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.917961877\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.917961877\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−9.660867534574481690205312493512, −9.225516466828060370524247894888, −8.539271092358702194391467963269, −8.257640973358794195307266579110, −7.80933020516206900900319067350, −7.26903454180533045084398071457, −6.68133765468582431283149091563, −6.06282765571084527336688583375, −5.29586506900316000330454330009, −5.11367386377376535827918890858, −4.39384328087896845067224792021, −3.60056369893769314029848916174, −3.07662112947169358105875225291, −1.86696234332261648050712199320, −1.25981186087362866088062482024,
1.25981186087362866088062482024, 1.86696234332261648050712199320, 3.07662112947169358105875225291, 3.60056369893769314029848916174, 4.39384328087896845067224792021, 5.11367386377376535827918890858, 5.29586506900316000330454330009, 6.06282765571084527336688583375, 6.68133765468582431283149091563, 7.26903454180533045084398071457, 7.80933020516206900900319067350, 8.257640973358794195307266579110, 8.539271092358702194391467963269, 9.225516466828060370524247894888, 9.660867534574481690205312493512