| L(s) = 1 | + 3-s − 2·9-s + 3·11-s + 3·19-s + 7·25-s − 5·27-s + 3·33-s − 15·41-s − 14·43-s + 6·49-s + 3·57-s − 3·59-s + 4·67-s − 3·73-s + 7·75-s + 81-s + 9·83-s − 3·89-s − 6·99-s − 27·107-s − 6·113-s − 9·121-s − 15·123-s + 127-s − 14·129-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2/3·9-s + 0.904·11-s + 0.688·19-s + 7/5·25-s − 0.962·27-s + 0.522·33-s − 2.34·41-s − 2.13·43-s + 6/7·49-s + 0.397·57-s − 0.390·59-s + 0.488·67-s − 0.351·73-s + 0.808·75-s + 1/9·81-s + 0.987·83-s − 0.317·89-s − 0.603·99-s − 2.61·107-s − 0.564·113-s − 0.818·121-s − 1.35·123-s + 0.0887·127-s − 1.23·129-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 43 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 107 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 60 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
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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
−7.39504258213978077112816112287, −6.88416322491043082376615293505, −6.64046483980678212658538961769, −6.34760448598166956215095113093, −5.57371725994754263415913002103, −5.28594706084533163961467791869, −4.92758795535568146025132522030, −4.31815259173784772592333214743, −3.72794607206376875268298257576, −3.35182995739394932719429753597, −2.99172841021428627016884109960, −2.38454915021009285600373075157, −1.67599693468812245466669737634, −1.14720251255688287004413686217, 0,
1.14720251255688287004413686217, 1.67599693468812245466669737634, 2.38454915021009285600373075157, 2.99172841021428627016884109960, 3.35182995739394932719429753597, 3.72794607206376875268298257576, 4.31815259173784772592333214743, 4.92758795535568146025132522030, 5.28594706084533163961467791869, 5.57371725994754263415913002103, 6.34760448598166956215095113093, 6.64046483980678212658538961769, 6.88416322491043082376615293505, 7.39504258213978077112816112287
