| L(s) = 1 | + 2·3-s − 4·5-s + 3·9-s − 4·11-s − 8·15-s + 6·25-s + 4·27-s − 8·31-s − 8·33-s + 4·37-s − 12·45-s + 16·47-s − 2·49-s − 4·53-s + 16·55-s − 8·59-s + 12·75-s + 5·81-s + 4·89-s − 16·93-s − 4·97-s − 12·99-s + 8·103-s + 8·111-s + 4·113-s + 5·121-s − 4·125-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.78·5-s + 9-s − 1.20·11-s − 2.06·15-s + 6/5·25-s + 0.769·27-s − 1.43·31-s − 1.39·33-s + 0.657·37-s − 1.78·45-s + 2.33·47-s − 2/7·49-s − 0.549·53-s + 2.15·55-s − 1.04·59-s + 1.38·75-s + 5/9·81-s + 0.423·89-s − 1.65·93-s − 0.406·97-s − 1.20·99-s + 0.788·103-s + 0.759·111-s + 0.376·113-s + 5/11·121-s − 0.357·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4460544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4460544 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 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
−7.39186319101629747751137410571, −7.05448628605400869728788710019, −6.51324664811269980005737780864, −5.84473708878542571500406397791, −5.53813754273097507935310957396, −4.89582312697205489813858923044, −4.50075075534743106849351021365, −4.10626880408486988099113971233, −3.71971995591812207975568195487, −3.26644027385238262999885670111, −2.89131281963204198409759454434, −2.32610423658807304356039324045, −1.78885533183265556015454713648, −0.812211549263806941495651919909, 0,
0.812211549263806941495651919909, 1.78885533183265556015454713648, 2.32610423658807304356039324045, 2.89131281963204198409759454434, 3.26644027385238262999885670111, 3.71971995591812207975568195487, 4.10626880408486988099113971233, 4.50075075534743106849351021365, 4.89582312697205489813858923044, 5.53813754273097507935310957396, 5.84473708878542571500406397791, 6.51324664811269980005737780864, 7.05448628605400869728788710019, 7.39186319101629747751137410571
