| L(s) = 1 | − 4-s + 16-s + 8·19-s + 18·29-s − 8·31-s − 12·41-s − 2·49-s − 2·61-s − 64-s + 24·71-s − 8·76-s + 32·79-s − 6·89-s + 12·101-s − 22·109-s − 18·116-s − 22·121-s + 8·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 12·164-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1/4·16-s + 1.83·19-s + 3.34·29-s − 1.43·31-s − 1.87·41-s − 2/7·49-s − 0.256·61-s − 1/8·64-s + 2.84·71-s − 0.917·76-s + 3.60·79-s − 0.635·89-s + 1.19·101-s − 2.10·109-s − 1.67·116-s − 2·121-s + 0.718·124-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 + 0.937·164-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16402500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16402500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.575988944\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.575988944\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−8.568898047053907878572376957310, −8.116066877666922280138122953717, −8.052227604401445531077485653725, −7.66960146563822627235602019293, −7.01268029498040075469767635507, −6.85895891960306442098594212485, −6.44285066467025587181054171199, −6.15985282688146409377175111943, −5.39834659508785558255424498877, −5.25821767358497785555253714993, −4.94375701648568252727273119844, −4.63673060107450965376658497772, −3.91698048710023497064247946627, −3.66355961316262205148517182955, −3.18236732006126239065697112126, −2.85068910397812692199959396366, −2.24789322602034699438186367725, −1.63088216983547463506067505073, −1.01922391665599618006517759318, −0.55821438895858008299415781681,
0.55821438895858008299415781681, 1.01922391665599618006517759318, 1.63088216983547463506067505073, 2.24789322602034699438186367725, 2.85068910397812692199959396366, 3.18236732006126239065697112126, 3.66355961316262205148517182955, 3.91698048710023497064247946627, 4.63673060107450965376658497772, 4.94375701648568252727273119844, 5.25821767358497785555253714993, 5.39834659508785558255424498877, 6.15985282688146409377175111943, 6.44285066467025587181054171199, 6.85895891960306442098594212485, 7.01268029498040075469767635507, 7.66960146563822627235602019293, 8.052227604401445531077485653725, 8.116066877666922280138122953717, 8.568898047053907878572376957310
