L(s) = 1 | + 4·9-s − 4·11-s + 8·19-s + 8·29-s − 4·31-s − 4·49-s − 8·59-s + 16·61-s − 4·71-s + 16·79-s + 7·81-s − 16·99-s + 4·101-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 8·169-s + 32·171-s + 173-s + 179-s + ⋯ |
L(s) = 1 | + 4/3·9-s − 1.20·11-s + 1.83·19-s + 1.48·29-s − 0.718·31-s − 4/7·49-s − 1.04·59-s + 2.04·61-s − 0.474·71-s + 1.80·79-s + 7/9·81-s − 1.60·99-s + 0.398·101-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-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 − 0.615·169-s + 2.44·171-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.405954606\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.405954606\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
good | 3 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 60 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 76 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 64 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 76 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
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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
−10.16295180583150965126291744346, −9.868969334000574295439406944488, −9.405693834573139579918885852563, −8.718300526206960332452318126453, −8.052337525827581049754872093080, −7.61941640722412707459745829344, −7.19690138581684033869632982521, −6.62247455175116184793468194379, −5.89286398287201986360553728178, −5.05067659183957414102306681851, −4.91777307744066443525508269587, −3.93610512047012510827116461824, −3.22993169807848032073755829234, −2.37995704665424455811000441929, −1.19648569880385905361105161627,
1.19648569880385905361105161627, 2.37995704665424455811000441929, 3.22993169807848032073755829234, 3.93610512047012510827116461824, 4.91777307744066443525508269587, 5.05067659183957414102306681851, 5.89286398287201986360553728178, 6.62247455175116184793468194379, 7.19690138581684033869632982521, 7.61941640722412707459745829344, 8.052337525827581049754872093080, 8.718300526206960332452318126453, 9.405693834573139579918885852563, 9.868969334000574295439406944488, 10.16295180583150965126291744346