| L(s) = 1 | − 2·5-s − 3·7-s + 9-s − 11-s + 3·13-s − 2·25-s − 11·31-s + 6·35-s + 4·43-s − 2·45-s − 12·47-s + 2·49-s + 2·55-s − 8·61-s − 3·63-s − 6·65-s − 10·67-s + 3·77-s − 8·81-s − 9·91-s − 99-s + 12·101-s − 4·103-s − 2·107-s + 10·113-s + 3·117-s − 9·121-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.13·7-s + 1/3·9-s − 0.301·11-s + 0.832·13-s − 2/5·25-s − 1.97·31-s + 1.01·35-s + 0.609·43-s − 0.298·45-s − 1.75·47-s + 2/7·49-s + 0.269·55-s − 1.02·61-s − 0.377·63-s − 0.744·65-s − 1.22·67-s + 0.341·77-s − 8/9·81-s − 0.943·91-s − 0.100·99-s + 1.19·101-s − 0.394·103-s − 0.193·107-s + 0.940·113-s + 0.277·117-s − 0.818·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62720 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62720 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 84 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 80 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 104 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 67 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 71 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
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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
−9.758800191054563874268148294615, −9.059474362162444084636653109102, −8.793596764633027673539306722802, −8.074990667663932006256916887755, −7.52078434616507789278169796029, −7.26250741159747012116594889457, −6.44331797321542749851386385772, −6.11446292148466110110284384420, −5.42407504893480721451111781069, −4.67663774575454162995432296310, −3.89266922650361393723442828180, −3.55686092469352041485825479634, −2.85564403443205593689686508796, −1.64828465259984423003421614545, 0,
1.64828465259984423003421614545, 2.85564403443205593689686508796, 3.55686092469352041485825479634, 3.89266922650361393723442828180, 4.67663774575454162995432296310, 5.42407504893480721451111781069, 6.11446292148466110110284384420, 6.44331797321542749851386385772, 7.26250741159747012116594889457, 7.52078434616507789278169796029, 8.074990667663932006256916887755, 8.793596764633027673539306722802, 9.059474362162444084636653109102, 9.758800191054563874268148294615
