| L(s) = 1 | + 3-s + 9-s − 3·11-s + 2·17-s − 8·19-s + 6·25-s + 27-s − 3·33-s + 7·41-s − 9·43-s − 5·49-s + 2·51-s − 8·57-s − 23·59-s + 7·67-s + 13·73-s + 6·75-s + 81-s + 19·83-s − 3·89-s − 2·97-s − 3·99-s + 7·107-s − 5·113-s − 3·121-s + 7·123-s + 127-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.904·11-s + 0.485·17-s − 1.83·19-s + 6/5·25-s + 0.192·27-s − 0.522·33-s + 1.09·41-s − 1.37·43-s − 5/7·49-s + 0.280·51-s − 1.05·57-s − 2.99·59-s + 0.855·67-s + 1.52·73-s + 0.692·75-s + 1/9·81-s + 2.08·83-s − 0.317·89-s − 0.203·97-s − 0.301·99-s + 0.676·107-s − 0.470·113-s − 0.272·121-s + 0.631·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1783296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1783296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 8 T + p T^{2} ) \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 9 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 112 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 9 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−7.83358910748423648257205248160, −7.16376391894635814087959727623, −6.79866427077655680229109324853, −6.28418883020590334772909027500, −6.06758220922893021476710814184, −5.30859801919420329839855171747, −4.89430022764209812025844516251, −4.57827498434006456118379582895, −4.01354881121351611063454741376, −3.35595623678010479206746130126, −3.05040555747868638302861071315, −2.34559092128197092484241727172, −1.97477029637729316409982379185, −1.09883264763357236202823260489, 0,
1.09883264763357236202823260489, 1.97477029637729316409982379185, 2.34559092128197092484241727172, 3.05040555747868638302861071315, 3.35595623678010479206746130126, 4.01354881121351611063454741376, 4.57827498434006456118379582895, 4.89430022764209812025844516251, 5.30859801919420329839855171747, 6.06758220922893021476710814184, 6.28418883020590334772909027500, 6.79866427077655680229109324853, 7.16376391894635814087959727623, 7.83358910748423648257205248160
