| L(s) = 1 | − 2·2-s + 3·4-s − 2·5-s − 4·8-s + 4·10-s + 2·11-s + 5·16-s − 6·17-s + 2·19-s − 6·20-s − 4·22-s − 4·23-s − 2·25-s + 10·31-s − 6·32-s + 12·34-s − 4·37-s − 4·38-s + 8·40-s − 6·41-s + 4·43-s + 6·44-s + 8·46-s − 6·47-s + 4·50-s + 12·53-s − 4·55-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.894·5-s − 1.41·8-s + 1.26·10-s + 0.603·11-s + 5/4·16-s − 1.45·17-s + 0.458·19-s − 1.34·20-s − 0.852·22-s − 0.834·23-s − 2/5·25-s + 1.79·31-s − 1.06·32-s + 2.05·34-s − 0.657·37-s − 0.648·38-s + 1.26·40-s − 0.937·41-s + 0.609·43-s + 0.904·44-s + 1.17·46-s − 0.875·47-s + 0.565·50-s + 1.64·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8638913744\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8638913744\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_4$ | \( 1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 10 T + 82 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 86 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 4 T - 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_4$ | \( 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 18 T + 182 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4 T - 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2 T + 162 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 162 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
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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
−7.84765487368768860769242742653, −7.71992016479634089860880991159, −7.15020599873679668348541086871, −6.90951080462553017174952159381, −6.62506979527121883220720456246, −6.43818126420135066311204821204, −5.93481120913551524054522139116, −5.54100819292567442174291173581, −5.23341744295641218995391737590, −4.51043541353326318563307519364, −4.35898993130214179007988660286, −4.07505504720581808236900605838, −3.37851601074741963049161761425, −3.32548423912170175310326166844, −2.59968979450742003780057019075, −2.37519692820945728285819048018, −1.69048504214662511733193828091, −1.54866508644472456882636244626, −0.67258874766466695522463288606, −0.39589012665285907675282373958,
0.39589012665285907675282373958, 0.67258874766466695522463288606, 1.54866508644472456882636244626, 1.69048504214662511733193828091, 2.37519692820945728285819048018, 2.59968979450742003780057019075, 3.32548423912170175310326166844, 3.37851601074741963049161761425, 4.07505504720581808236900605838, 4.35898993130214179007988660286, 4.51043541353326318563307519364, 5.23341744295641218995391737590, 5.54100819292567442174291173581, 5.93481120913551524054522139116, 6.43818126420135066311204821204, 6.62506979527121883220720456246, 6.90951080462553017174952159381, 7.15020599873679668348541086871, 7.71992016479634089860880991159, 7.84765487368768860769242742653
