| L(s) = 1 | − 2·7-s − 9-s − 8·17-s + 4·23-s − 6·25-s − 24·47-s + 3·49-s + 2·63-s + 28·71-s + 4·73-s − 24·79-s + 81-s − 4·97-s − 32·103-s + 20·113-s + 16·119-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 8·153-s + 157-s − 8·161-s + 163-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 1/3·9-s − 1.94·17-s + 0.834·23-s − 6/5·25-s − 3.50·47-s + 3/7·49-s + 0.251·63-s + 3.32·71-s + 0.468·73-s − 2.70·79-s + 1/9·81-s − 0.406·97-s − 3.15·103-s + 1.88·113-s + 1.46·119-s + 1.63·121-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 + 0.646·153-s + 0.0798·157-s − 0.630·161-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28901376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28901376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−8.116851159532703854246143445539, −7.70834149207927676619920451311, −7.09611571090695418765319034525, −6.85782672950787220939322583308, −6.67616073900524526108472432769, −6.21294747421646970658788295130, −5.99617407797385932841893441289, −5.43515134825974356666601140719, −5.02322372623098884776868496175, −4.81710203151857549674279171297, −4.16101884963906831827577296676, −4.02405480876161718604589134660, −3.34378273326167599288314235215, −3.17807122904881126005244579890, −2.55507776657246030308401610618, −2.19219352449159311691295553007, −1.71958472548994670981627071126, −1.05352039163140314541465274874, 0, 0,
1.05352039163140314541465274874, 1.71958472548994670981627071126, 2.19219352449159311691295553007, 2.55507776657246030308401610618, 3.17807122904881126005244579890, 3.34378273326167599288314235215, 4.02405480876161718604589134660, 4.16101884963906831827577296676, 4.81710203151857549674279171297, 5.02322372623098884776868496175, 5.43515134825974356666601140719, 5.99617407797385932841893441289, 6.21294747421646970658788295130, 6.67616073900524526108472432769, 6.85782672950787220939322583308, 7.09611571090695418765319034525, 7.70834149207927676619920451311, 8.116851159532703854246143445539
