| L(s) = 1 | + 4·5-s − 9-s + 8·11-s + 4·19-s + 11·25-s − 12·29-s + 12·31-s − 4·45-s − 49-s + 32·55-s − 8·59-s − 4·61-s − 16·71-s − 32·79-s + 81-s − 32·89-s + 16·95-s − 8·99-s + 16·101-s − 36·109-s + 26·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s − 48·145-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 1/3·9-s + 2.41·11-s + 0.917·19-s + 11/5·25-s − 2.22·29-s + 2.15·31-s − 0.596·45-s − 1/7·49-s + 4.31·55-s − 1.04·59-s − 0.512·61-s − 1.89·71-s − 3.60·79-s + 1/9·81-s − 3.39·89-s + 1.64·95-s − 0.804·99-s + 1.59·101-s − 3.44·109-s + 2.36·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3.98·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.760969972\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.760969972\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 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
−11.24722093977470854702683840386, −11.24019793743423503275799088355, −10.18195855850965947459725594849, −10.15108226649660678129842973681, −9.400540590551782899011016515411, −9.373846238631850193035615484065, −8.880243444565989468907626311507, −8.456344019713605042974294084634, −7.66657305421981843105497912975, −7.06270346737010173050421499299, −6.64186150562658124434502543287, −6.23716413693100665806605864196, −5.59208663617913350095037933104, −5.57745094941215196916442241126, −4.46122508475892966967188002404, −4.17941370642037650400736746198, −3.19954430820490038521196244837, −2.72924911726652814890585679813, −1.57802764791372544356520025922, −1.39269080588721714232351349370,
1.39269080588721714232351349370, 1.57802764791372544356520025922, 2.72924911726652814890585679813, 3.19954430820490038521196244837, 4.17941370642037650400736746198, 4.46122508475892966967188002404, 5.57745094941215196916442241126, 5.59208663617913350095037933104, 6.23716413693100665806605864196, 6.64186150562658124434502543287, 7.06270346737010173050421499299, 7.66657305421981843105497912975, 8.456344019713605042974294084634, 8.880243444565989468907626311507, 9.373846238631850193035615484065, 9.400540590551782899011016515411, 10.15108226649660678129842973681, 10.18195855850965947459725594849, 11.24019793743423503275799088355, 11.24722093977470854702683840386
