| L(s) = 1 | + 4-s + 8·5-s − 3·9-s + 4·11-s − 4·13-s + 16-s − 4·17-s + 8·20-s + 23-s + 38·25-s − 3·36-s + 4·44-s − 24·45-s + 2·49-s − 4·52-s − 8·53-s + 32·55-s + 64-s − 32·65-s − 4·68-s + 12·73-s + 8·80-s + 9·81-s + 28·83-s − 32·85-s − 12·89-s + 92-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 3.57·5-s − 9-s + 1.20·11-s − 1.10·13-s + 1/4·16-s − 0.970·17-s + 1.78·20-s + 0.208·23-s + 38/5·25-s − 1/2·36-s + 0.603·44-s − 3.57·45-s + 2/7·49-s − 0.554·52-s − 1.09·53-s + 4.31·55-s + 1/8·64-s − 3.96·65-s − 0.485·68-s + 1.40·73-s + 0.894·80-s + 81-s + 3.07·83-s − 3.47·85-s − 1.27·89-s + 0.104·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 438012 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 438012 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.206707818\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.206707818\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 23 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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
−8.971935452602447588346290189052, −8.300196560171154480855065910688, −7.62735421295774645335638863042, −6.69441913641732873591204403077, −6.69265944578033613099294080985, −6.35362788354985461499174366441, −5.84432417959727908710252168629, −5.35356885744180803495724283233, −5.12924034341750180180238442817, −4.47212152404542749707456940527, −3.42573528437023625765419608105, −2.63232737474552247848840602025, −2.38695715760664154900264131006, −1.87977103445283065955695773340, −1.21768411761699883032703079663,
1.21768411761699883032703079663, 1.87977103445283065955695773340, 2.38695715760664154900264131006, 2.63232737474552247848840602025, 3.42573528437023625765419608105, 4.47212152404542749707456940527, 5.12924034341750180180238442817, 5.35356885744180803495724283233, 5.84432417959727908710252168629, 6.35362788354985461499174366441, 6.69265944578033613099294080985, 6.69441913641732873591204403077, 7.62735421295774645335638863042, 8.300196560171154480855065910688, 8.971935452602447588346290189052
