| L(s) = 1 | − 4·5-s + 4·13-s + 8·25-s − 20·29-s − 12·37-s + 28·49-s + 4·53-s + 36·61-s − 16·65-s − 81-s + 32·97-s + 4·101-s + 28·109-s − 56·113-s − 28·125-s + 127-s + 131-s + 137-s + 139-s + 80·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 1.10·13-s + 8/5·25-s − 3.71·29-s − 1.97·37-s + 4·49-s + 0.549·53-s + 4.60·61-s − 1.98·65-s − 1/9·81-s + 3.24·97-s + 0.398·101-s + 2.68·109-s − 5.26·113-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 6.64·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 8/13·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.664142674\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.664142674\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 11 | $C_2^3$ | \( 1 - 206 T^{4} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2^3$ | \( 1 - 238 T^{4} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 43 | $C_2^3$ | \( 1 - 334 T^{4} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 + 3442 T^{4} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 4946 T^{4} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 13294 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.90202629715169034192104344324, −6.73244162201691129730565405523, −6.25802322234281062037752543160, −6.17007651870074428249090510843, −5.64861350986827531154328150181, −5.60072681486752430754928570444, −5.49811604256273294468689325286, −5.30948618090500992178831313295, −5.12339961833765149524974880566, −4.71648230017184158761813836350, −4.29916545727762915007225748763, −4.19747737108337577759961088476, −3.90511098981041082547290582441, −3.81141659169941157034012869332, −3.74020931537465663617099305837, −3.48228695733909583209286092204, −3.09841844138433145637407699531, −2.94282362685927355379220388573, −2.37354882939843205552529058327, −2.19137560522569391855279020432, −1.83027844639495985367950262236, −1.75472567687767477818837934608, −0.909458151776791470990038601548, −0.796942670397594155387968528835, −0.34181272075874874061229082379,
0.34181272075874874061229082379, 0.796942670397594155387968528835, 0.909458151776791470990038601548, 1.75472567687767477818837934608, 1.83027844639495985367950262236, 2.19137560522569391855279020432, 2.37354882939843205552529058327, 2.94282362685927355379220388573, 3.09841844138433145637407699531, 3.48228695733909583209286092204, 3.74020931537465663617099305837, 3.81141659169941157034012869332, 3.90511098981041082547290582441, 4.19747737108337577759961088476, 4.29916545727762915007225748763, 4.71648230017184158761813836350, 5.12339961833765149524974880566, 5.30948618090500992178831313295, 5.49811604256273294468689325286, 5.60072681486752430754928570444, 5.64861350986827531154328150181, 6.17007651870074428249090510843, 6.25802322234281062037752543160, 6.73244162201691129730565405523, 6.90202629715169034192104344324
