| L(s) = 1 | − 7-s + 9-s − 6·11-s − 6·23-s + 2·25-s + 4·29-s − 4·43-s + 49-s + 4·53-s − 63-s − 8·67-s − 6·71-s + 6·77-s + 16·79-s + 81-s − 6·99-s + 6·107-s − 8·109-s + 4·113-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 1/3·9-s − 1.80·11-s − 1.25·23-s + 2/5·25-s + 0.742·29-s − 0.609·43-s + 1/7·49-s + 0.549·53-s − 0.125·63-s − 0.977·67-s − 0.712·71-s + 0.683·77-s + 1.80·79-s + 1/9·81-s − 0.603·99-s + 0.580·107-s − 0.766·109-s + 0.376·113-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.155465865\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.155465865\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 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
−8.173119478204142610657905117257, −7.75794185837280279181031695245, −7.55678486819680889512552808838, −6.85220759875859176644246411749, −6.54066007710274840220065963461, −5.96639596496664211673275869411, −5.51589290946902185965895504532, −5.08597086802524050952535612493, −4.58010161848683744330249234523, −4.07778719600358940995665772920, −3.40743049026903111579256345871, −2.86050294477403871935983467054, −2.37282732521177382928381023214, −1.66596272251178672985474906255, −0.50494018450613813830501897510,
0.50494018450613813830501897510, 1.66596272251178672985474906255, 2.37282732521177382928381023214, 2.86050294477403871935983467054, 3.40743049026903111579256345871, 4.07778719600358940995665772920, 4.58010161848683744330249234523, 5.08597086802524050952535612493, 5.51589290946902185965895504532, 5.96639596496664211673275869411, 6.54066007710274840220065963461, 6.85220759875859176644246411749, 7.55678486819680889512552808838, 7.75794185837280279181031695245, 8.173119478204142610657905117257
