| L(s) = 1 | + 4·3-s + 4·7-s + 6·9-s + 4·19-s + 16·21-s − 2·25-s − 4·27-s − 12·29-s + 16·31-s + 4·37-s + 9·49-s − 12·53-s + 16·57-s + 12·59-s + 24·63-s − 8·75-s − 37·81-s − 12·83-s − 48·87-s + 64·93-s − 8·103-s − 28·109-s + 16·111-s − 36·113-s + 10·121-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 1.51·7-s + 2·9-s + 0.917·19-s + 3.49·21-s − 2/5·25-s − 0.769·27-s − 2.22·29-s + 2.87·31-s + 0.657·37-s + 9/7·49-s − 1.64·53-s + 2.11·57-s + 1.56·59-s + 3.02·63-s − 0.923·75-s − 4.11·81-s − 1.31·83-s − 5.14·87-s + 6.63·93-s − 0.788·103-s − 2.68·109-s + 1.51·111-s − 3.38·113-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.343233022\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.343233022\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−11.25706804491268867726777168842, −11.01287155568625878927977115204, −10.26292381518721065546968140955, −9.588953830211663260248387217318, −9.550059819250600182882265266760, −9.036128236832154519105890323137, −8.345910353654609925558497678554, −8.247623890924368214748670444184, −7.907070679167903386448456167174, −7.52496880825596917724798965123, −6.98708042166248761297055838228, −6.16478904905507191535804434872, −5.50244752816487965820653538501, −5.08387460991897900681181111709, −4.17067380789998926272961730796, −3.99748740106848241050424196853, −3.08433232669730184582890431382, −2.73617141151098625327292214363, −2.05462471309512005043083227882, −1.40351027887794056703929038807,
1.40351027887794056703929038807, 2.05462471309512005043083227882, 2.73617141151098625327292214363, 3.08433232669730184582890431382, 3.99748740106848241050424196853, 4.17067380789998926272961730796, 5.08387460991897900681181111709, 5.50244752816487965820653538501, 6.16478904905507191535804434872, 6.98708042166248761297055838228, 7.52496880825596917724798965123, 7.907070679167903386448456167174, 8.247623890924368214748670444184, 8.345910353654609925558497678554, 9.036128236832154519105890323137, 9.550059819250600182882265266760, 9.588953830211663260248387217318, 10.26292381518721065546968140955, 11.01287155568625878927977115204, 11.25706804491268867726777168842
