| L(s) = 1 | + 4·5-s + 9-s − 2·13-s + 2·25-s − 18·29-s + 10·37-s − 16·41-s + 4·45-s − 4·49-s + 6·53-s − 6·61-s − 8·65-s + 4·73-s + 81-s + 8·89-s − 12·97-s − 12·101-s + 4·109-s + 8·113-s − 2·117-s + 18·121-s − 28·125-s + 127-s + 131-s + 137-s + 139-s − 72·145-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 1/3·9-s − 0.554·13-s + 2/5·25-s − 3.34·29-s + 1.64·37-s − 2.49·41-s + 0.596·45-s − 4/7·49-s + 0.824·53-s − 0.768·61-s − 0.992·65-s + 0.468·73-s + 1/9·81-s + 0.847·89-s − 1.21·97-s − 1.19·101-s + 0.383·109-s + 0.752·113-s − 0.184·117-s + 1.63·121-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5.97·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331712 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331712 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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$ | \( ( 1 - T )( 1 + T ) \) |
| 17 | $C_2$ | \( 1 + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 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
−7.76624713192917086737434019109, −7.17451883464945116042481642099, −7.00554197718583441720853129117, −6.19639076899508835863027782995, −6.07670076607522553853918131621, −5.58993812523657640733256787134, −5.17723177206823965275365447249, −4.82254184615917527814345967487, −3.99349311314655594972590946465, −3.68751515630998838410085092260, −2.94100364208166329395153170087, −2.21314308085278224589908036658, −1.92355715876634755709631929622, −1.39474613990019727812604011373, 0,
1.39474613990019727812604011373, 1.92355715876634755709631929622, 2.21314308085278224589908036658, 2.94100364208166329395153170087, 3.68751515630998838410085092260, 3.99349311314655594972590946465, 4.82254184615917527814345967487, 5.17723177206823965275365447249, 5.58993812523657640733256787134, 6.07670076607522553853918131621, 6.19639076899508835863027782995, 7.00554197718583441720853129117, 7.17451883464945116042481642099, 7.76624713192917086737434019109
