| L(s) = 1 | + 4-s + 2·7-s + 9-s + 2·11-s − 6·13-s + 16-s + 2·17-s + 25-s + 2·28-s + 10·29-s + 36-s + 4·37-s + 10·43-s + 2·44-s − 6·47-s − 2·49-s − 6·52-s − 12·53-s + 2·59-s + 2·63-s + 64-s + 2·68-s + 4·77-s + 81-s + 14·89-s − 12·91-s + 2·97-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 0.755·7-s + 1/3·9-s + 0.603·11-s − 1.66·13-s + 1/4·16-s + 0.485·17-s + 1/5·25-s + 0.377·28-s + 1.85·29-s + 1/6·36-s + 0.657·37-s + 1.52·43-s + 0.301·44-s − 0.875·47-s − 2/7·49-s − 0.832·52-s − 1.64·53-s + 0.260·59-s + 0.251·63-s + 1/8·64-s + 0.242·68-s + 0.455·77-s + 1/9·81-s + 1.48·89-s − 1.25·91-s + 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 101124 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101124 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.952282511\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.952282511\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 53 | $C_2$ | \( 1 + 12 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 21 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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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
−9.539917923517974894193702439745, −9.206046778185703565602462746293, −8.512583931038736495994142781477, −7.87875582614648417893410130364, −7.70485273044520746179596079097, −7.08774059649357153394251333984, −6.56239168560119507277461437061, −6.11592303718561477997729268735, −5.34385401185269491278379000878, −4.64619350452250687221500821178, −4.56440027603299342495637618296, −3.50105508954324540065708393631, −2.76737442624040199291157728613, −2.09827210699150975251335021222, −1.12020746580009515163350049187,
1.12020746580009515163350049187, 2.09827210699150975251335021222, 2.76737442624040199291157728613, 3.50105508954324540065708393631, 4.56440027603299342495637618296, 4.64619350452250687221500821178, 5.34385401185269491278379000878, 6.11592303718561477997729268735, 6.56239168560119507277461437061, 7.08774059649357153394251333984, 7.70485273044520746179596079097, 7.87875582614648417893410130364, 8.512583931038736495994142781477, 9.206046778185703565602462746293, 9.539917923517974894193702439745
