| L(s) = 1 | + 4·5-s + 5·9-s + 2·11-s + 2·19-s + 11·25-s + 2·29-s + 2·31-s + 20·45-s + 13·49-s + 8·55-s + 8·59-s − 14·61-s − 10·71-s + 8·79-s + 16·81-s + 14·89-s + 8·95-s + 10·99-s − 20·101-s − 20·109-s + 3·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 5/3·9-s + 0.603·11-s + 0.458·19-s + 11/5·25-s + 0.371·29-s + 0.359·31-s + 2.98·45-s + 13/7·49-s + 1.07·55-s + 1.04·59-s − 1.79·61-s − 1.18·71-s + 0.900·79-s + 16/9·81-s + 1.48·89-s + 0.820·95-s + 1.00·99-s − 1.99·101-s − 1.91·109-s + 3/11·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.664·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 774400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 774400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.000047326\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.000047326\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 33 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−10.13898238215407571663734279985, −10.07564368925042555799865877887, −9.403301977539821374141066299398, −9.313045457277553068622508299338, −8.897743975286703712645999652806, −8.323655504051748825063387532869, −7.68351829663461662789449949694, −7.32222619783932134211396301169, −6.70993892708165864018464839749, −6.65626184671590531540661971855, −5.93158406856047361028501662416, −5.72286456632019030961288228297, −4.90208226681841181061947190100, −4.79098303196483030742787910605, −4.01156032124327073947183754904, −3.60471629920815387070579509162, −2.64608944348735857152053018683, −2.30114734216782872578266309561, −1.35150114447800051738599593194, −1.23609497013526401921280739250,
1.23609497013526401921280739250, 1.35150114447800051738599593194, 2.30114734216782872578266309561, 2.64608944348735857152053018683, 3.60471629920815387070579509162, 4.01156032124327073947183754904, 4.79098303196483030742787910605, 4.90208226681841181061947190100, 5.72286456632019030961288228297, 5.93158406856047361028501662416, 6.65626184671590531540661971855, 6.70993892708165864018464839749, 7.32222619783932134211396301169, 7.68351829663461662789449949694, 8.323655504051748825063387532869, 8.897743975286703712645999652806, 9.313045457277553068622508299338, 9.403301977539821374141066299398, 10.07564368925042555799865877887, 10.13898238215407571663734279985
