| L(s) = 1 | − 3·3-s + 5-s − 7-s + 6·9-s + 6·11-s − 2·13-s − 3·15-s + 8·19-s + 3·21-s + 9·23-s − 9·27-s − 3·29-s − 4·31-s − 18·33-s − 35-s + 16·37-s + 6·39-s + 3·41-s + 8·43-s + 6·45-s − 3·47-s + 7·49-s + 12·53-s + 6·55-s − 24·57-s + 6·59-s + 13·61-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.447·5-s − 0.377·7-s + 2·9-s + 1.80·11-s − 0.554·13-s − 0.774·15-s + 1.83·19-s + 0.654·21-s + 1.87·23-s − 1.73·27-s − 0.557·29-s − 0.718·31-s − 3.13·33-s − 0.169·35-s + 2.63·37-s + 0.960·39-s + 0.468·41-s + 1.21·43-s + 0.894·45-s − 0.437·47-s + 49-s + 1.64·53-s + 0.809·55-s − 3.17·57-s + 0.781·59-s + 1.66·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.508393540\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.508393540\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 13 T + 102 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 10 T + 21 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−10.62699894040910626305379731916, −10.27721825181985521898558990473, −9.677358948156739509564546648466, −9.465929525481541844971763147613, −9.170002270595554644066441672151, −8.723714346463893322233000053260, −7.75767531866467805113213700477, −7.34788386273809706028260722473, −6.97418829706202091809810403279, −6.75031775607795694985892920470, −5.99429893171094008643750801277, −5.80295349566630215202399716279, −5.37606225425339136159483735082, −4.85283731608582176135917784889, −4.22769782145272350158919093450, −3.86912304158204033681038489741, −3.05802772614047974712709437189, −2.30275679563339956255812454951, −1.05214766785694222593918158316, −1.00139764272227101364269385792,
1.00139764272227101364269385792, 1.05214766785694222593918158316, 2.30275679563339956255812454951, 3.05802772614047974712709437189, 3.86912304158204033681038489741, 4.22769782145272350158919093450, 4.85283731608582176135917784889, 5.37606225425339136159483735082, 5.80295349566630215202399716279, 5.99429893171094008643750801277, 6.75031775607795694985892920470, 6.97418829706202091809810403279, 7.34788386273809706028260722473, 7.75767531866467805113213700477, 8.723714346463893322233000053260, 9.170002270595554644066441672151, 9.465929525481541844971763147613, 9.677358948156739509564546648466, 10.27721825181985521898558990473, 10.62699894040910626305379731916
