| L(s) = 1 | + 4·5-s − 6·7-s − 9-s − 2·11-s + 2·17-s + 2·19-s − 10·23-s + 11·25-s − 10·29-s − 24·35-s + 8·37-s + 20·43-s − 4·45-s − 6·47-s + 18·49-s − 8·55-s − 10·59-s + 10·61-s + 6·63-s − 4·67-s + 32·71-s − 18·73-s + 12·77-s − 32·79-s + 81-s + 8·85-s − 12·89-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 2.26·7-s − 1/3·9-s − 0.603·11-s + 0.485·17-s + 0.458·19-s − 2.08·23-s + 11/5·25-s − 1.85·29-s − 4.05·35-s + 1.31·37-s + 3.04·43-s − 0.596·45-s − 0.875·47-s + 18/7·49-s − 1.07·55-s − 1.30·59-s + 1.28·61-s + 0.755·63-s − 0.488·67-s + 3.79·71-s − 2.10·73-s + 1.36·77-s − 3.60·79-s + 1/9·81-s + 0.867·85-s − 1.27·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3686400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3686400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.392414317\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.392414317\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−9.696574835597063382283682651580, −9.038549955252746710724201379352, −8.953802812935536025516489786454, −8.267354219491859696791336348508, −7.74576369844475019623970469419, −7.33316000422891545983308457418, −7.09328949726502676596542514748, −6.32069259284422154173722212054, −6.12652013082554136850736304366, −5.90432491861134643142902743445, −5.67209577824284190143846731144, −5.18985888663582325983209216170, −4.46055482596406586374859610340, −3.90158321036534156877060658250, −3.49183257415833370224324902038, −2.91595503496291100763378625931, −2.52134058101817711327575824664, −2.17363045451128222286711996109, −1.37680638139699943414809539275, −0.41961400063944562489264206163,
0.41961400063944562489264206163, 1.37680638139699943414809539275, 2.17363045451128222286711996109, 2.52134058101817711327575824664, 2.91595503496291100763378625931, 3.49183257415833370224324902038, 3.90158321036534156877060658250, 4.46055482596406586374859610340, 5.18985888663582325983209216170, 5.67209577824284190143846731144, 5.90432491861134643142902743445, 6.12652013082554136850736304366, 6.32069259284422154173722212054, 7.09328949726502676596542514748, 7.33316000422891545983308457418, 7.74576369844475019623970469419, 8.267354219491859696791336348508, 8.953802812935536025516489786454, 9.038549955252746710724201379352, 9.696574835597063382283682651580
