| L(s) = 1 | − 2·5-s + 16·19-s − 25-s − 12·29-s − 16·31-s − 12·41-s − 2·49-s − 12·61-s + 32·71-s + 16·79-s − 20·89-s − 32·95-s − 28·101-s + 20·109-s − 22·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s + 24·145-s + 149-s + 151-s + 32·155-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 3.67·19-s − 1/5·25-s − 2.22·29-s − 2.87·31-s − 1.87·41-s − 2/7·49-s − 1.53·61-s + 3.79·71-s + 1.80·79-s − 2.11·89-s − 3.28·95-s − 2.78·101-s + 1.91·109-s − 2·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.99·145-s + 0.0819·149-s + 0.0813·151-s + 2.57·155-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.212335175\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.212335175\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 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
−9.610342966639099694925149433255, −9.333477879231934759220167449011, −9.226058171897001814659227291784, −8.265117223089157239411512886960, −8.209151443680084889922028916982, −7.52342135744687534022378169885, −7.41144368419041654986731398540, −7.13402074126162639589090427692, −6.64504622952495083972104400827, −5.79360653219560157313105291554, −5.57248287375360996119390763337, −5.10987422315487353294961914387, −4.96621721210875437351677184952, −3.85664316549330218630053728810, −3.71599304262311115499967452216, −3.41851728508801929838304482548, −2.85350458962345062408189968866, −1.87322452308456859788832528505, −1.49057005351059371121255204327, −0.45336944881666730517742830952,
0.45336944881666730517742830952, 1.49057005351059371121255204327, 1.87322452308456859788832528505, 2.85350458962345062408189968866, 3.41851728508801929838304482548, 3.71599304262311115499967452216, 3.85664316549330218630053728810, 4.96621721210875437351677184952, 5.10987422315487353294961914387, 5.57248287375360996119390763337, 5.79360653219560157313105291554, 6.64504622952495083972104400827, 7.13402074126162639589090427692, 7.41144368419041654986731398540, 7.52342135744687534022378169885, 8.209151443680084889922028916982, 8.265117223089157239411512886960, 9.226058171897001814659227291784, 9.333477879231934759220167449011, 9.610342966639099694925149433255
