| L(s) = 1 | − 2·5-s + 9-s − 4·11-s + 3·25-s + 8·31-s + 4·37-s − 2·45-s + 6·49-s − 12·53-s + 8·55-s − 8·59-s + 81-s − 20·89-s + 4·97-s − 4·99-s + 4·113-s + 5·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 16·155-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1/3·9-s − 1.20·11-s + 3/5·25-s + 1.43·31-s + 0.657·37-s − 0.298·45-s + 6/7·49-s − 1.64·53-s + 1.07·55-s − 1.04·59-s + 1/9·81-s − 2.11·89-s + 0.406·97-s − 0.402·99-s + 0.376·113-s + 5/11·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 1.28·155-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 126 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−7.81459338329316092456303471934, −7.18119771671535295742271888233, −6.91482484640042720635426009826, −6.31164284230592953988010966290, −5.94099986968685787851757697385, −5.39781962658243035800899427040, −4.84017858258343887127501790738, −4.56327669906052856100427898935, −4.10004384681450319932940975163, −3.51212077472804641861565965606, −2.87823642369200791415665019044, −2.64342898436827693446784139701, −1.76110497281900272151880383100, −0.931483473849049737723533286704, 0,
0.931483473849049737723533286704, 1.76110497281900272151880383100, 2.64342898436827693446784139701, 2.87823642369200791415665019044, 3.51212077472804641861565965606, 4.10004384681450319932940975163, 4.56327669906052856100427898935, 4.84017858258343887127501790738, 5.39781962658243035800899427040, 5.94099986968685787851757697385, 6.31164284230592953988010966290, 6.91482484640042720635426009826, 7.18119771671535295742271888233, 7.81459338329316092456303471934
