| L(s) = 1 | + 5-s + 9-s + 5·13-s − 2·17-s − 4·25-s − 29-s − 7·37-s + 2·41-s + 45-s − 11·49-s − 14·53-s − 23·61-s + 5·65-s + 4·73-s + 81-s − 2·85-s − 23·89-s − 4·97-s + 3·101-s + 12·109-s − 10·113-s + 5·117-s − 4·121-s − 9·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1/3·9-s + 1.38·13-s − 0.485·17-s − 4/5·25-s − 0.185·29-s − 1.15·37-s + 0.312·41-s + 0.149·45-s − 1.57·49-s − 1.92·53-s − 2.94·61-s + 0.620·65-s + 0.468·73-s + 1/9·81-s − 0.216·85-s − 2.43·89-s − 0.406·97-s + 0.298·101-s + 1.14·109-s − 0.940·113-s + 0.462·117-s − 0.363·121-s − 0.804·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 59 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 49 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 93 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 112 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 9 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.974605038747319260347899511734, −7.62329133430492851707762764069, −6.91393910159089662776991507505, −6.63564833954828715360791632064, −6.11174492761105123074092884290, −5.83838510961250979724115036273, −5.30733020559734190685205547718, −4.56670313338449658938653928914, −4.41959366784876658772613758243, −3.53371472535671358054267168031, −3.33691532423917697990781264876, −2.54286807681346644140127916420, −1.65641967415603678097228744396, −1.45091644155074397942301651350, 0,
1.45091644155074397942301651350, 1.65641967415603678097228744396, 2.54286807681346644140127916420, 3.33691532423917697990781264876, 3.53371472535671358054267168031, 4.41959366784876658772613758243, 4.56670313338449658938653928914, 5.30733020559734190685205547718, 5.83838510961250979724115036273, 6.11174492761105123074092884290, 6.63564833954828715360791632064, 6.91393910159089662776991507505, 7.62329133430492851707762764069, 7.974605038747319260347899511734
