| L(s) = 1 | + 8·5-s + 39·25-s + 80·29-s + 160·41-s − 98·49-s + 44·61-s + 320·89-s + 80·101-s − 364·109-s + 242·121-s + 112·125-s + 127-s + 131-s + 137-s + 139-s + 640·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 238·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 8/5·5-s + 1.55·25-s + 2.75·29-s + 3.90·41-s − 2·49-s + 0.721·61-s + 3.59·89-s + 0.792·101-s − 3.33·109-s + 2·121-s + 0.895·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 4.41·145-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 1.40·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.254341772\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.254341772\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 8 T + p^{2} T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )( 1 + 10 T + p^{2} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )( 1 + 16 T + p^{2} T^{2} ) \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 40 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 70 T + p^{2} T^{2} )( 1 + 70 T + p^{2} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 80 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 56 T + p^{2} T^{2} )( 1 + 56 T + p^{2} T^{2} ) \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 110 T + p^{2} T^{2} )( 1 + 110 T + p^{2} T^{2} ) \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 160 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 130 T + p^{2} T^{2} )( 1 + 130 T + p^{2} T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.25897143868498582204806844618, −10.06209609593778240058757712816, −9.509845417310531378175017399470, −9.291737438529966952632907849883, −8.807950454077778297592798289202, −8.352699868080623691552539794036, −7.79384236081318026139011622890, −7.46890873056342618583782140776, −6.57926182402594870502176612674, −6.50604283296682602516088397348, −6.02747037150496146758744486226, −5.61980897062721922562137901515, −4.85989479768828035598562976539, −4.74682511606462404615535684663, −3.99152190107949918441622133530, −3.20685489441831752310022918616, −2.53831365851164639560552336411, −2.31722429720029262513114357722, −1.33381644013624857409242843241, −0.797299987566476688261041837917,
0.797299987566476688261041837917, 1.33381644013624857409242843241, 2.31722429720029262513114357722, 2.53831365851164639560552336411, 3.20685489441831752310022918616, 3.99152190107949918441622133530, 4.74682511606462404615535684663, 4.85989479768828035598562976539, 5.61980897062721922562137901515, 6.02747037150496146758744486226, 6.50604283296682602516088397348, 6.57926182402594870502176612674, 7.46890873056342618583782140776, 7.79384236081318026139011622890, 8.352699868080623691552539794036, 8.807950454077778297592798289202, 9.291737438529966952632907849883, 9.509845417310531378175017399470, 10.06209609593778240058757712816, 10.25897143868498582204806844618
