| L(s) = 1 | − 2·5-s + 2·9-s + 4·11-s − 4·19-s − 25-s − 4·29-s + 16·31-s + 4·41-s − 4·45-s − 6·49-s − 8·55-s + 20·59-s + 12·61-s − 8·71-s − 5·81-s + 4·89-s + 8·95-s + 8·99-s + 12·101-s − 20·109-s − 6·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 2/3·9-s + 1.20·11-s − 0.917·19-s − 1/5·25-s − 0.742·29-s + 2.87·31-s + 0.624·41-s − 0.596·45-s − 6/7·49-s − 1.07·55-s + 2.60·59-s + 1.53·61-s − 0.949·71-s − 5/9·81-s + 0.423·89-s + 0.820·95-s + 0.804·99-s + 1.19·101-s − 1.91·109-s − 0.545·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.664·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.411798966\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.411798966\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 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
−9.632090527523456196909016812554, −9.016211413026599097899843019882, −8.452787750824036051955566239342, −8.175981257035596164933228564207, −7.60666149794331837080893335181, −6.96141922523518447328239093096, −6.63533584028344857461098541140, −6.14899175557928899769675490070, −5.40030754931205132427766412299, −4.59526958741361978754262342630, −4.13919056539703828339818577688, −3.84344893939442439211466318812, −2.93119217667177575931323533226, −2.01470677816458276958706728259, −0.922315328074077157916315159630,
0.922315328074077157916315159630, 2.01470677816458276958706728259, 2.93119217667177575931323533226, 3.84344893939442439211466318812, 4.13919056539703828339818577688, 4.59526958741361978754262342630, 5.40030754931205132427766412299, 6.14899175557928899769675490070, 6.63533584028344857461098541140, 6.96141922523518447328239093096, 7.60666149794331837080893335181, 8.175981257035596164933228564207, 8.452787750824036051955566239342, 9.016211413026599097899843019882, 9.632090527523456196909016812554
