| L(s) = 1 | − 3·5-s + 4·7-s + 9-s + 2·13-s + 4·25-s − 12·35-s + 2·37-s − 3·45-s + 4·47-s − 49-s − 8·61-s + 4·63-s − 6·65-s + 16·67-s − 20·73-s + 8·79-s − 8·81-s + 24·83-s + 8·91-s + 8·97-s + 16·101-s + 2·117-s + 10·121-s + 3·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 1.51·7-s + 1/3·9-s + 0.554·13-s + 4/5·25-s − 2.02·35-s + 0.328·37-s − 0.447·45-s + 0.583·47-s − 1/7·49-s − 1.02·61-s + 0.503·63-s − 0.744·65-s + 1.95·67-s − 2.34·73-s + 0.900·79-s − 8/9·81-s + 2.63·83-s + 0.838·91-s + 0.812·97-s + 1.59·101-s + 0.184·117-s + 0.909·121-s + 0.268·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 540800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.798599779\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.798599779\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
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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
−8.455818670319071853073892193518, −7.968590726590330501123767305785, −7.64606590933620344425663891374, −7.32469371722216245826150993062, −6.76630124393435907765378699325, −6.17938178595195748928406873146, −5.67339607138908642684889275430, −5.00183036684168281129084284042, −4.59037596784972644267108292819, −4.31690762898032664126534932318, −3.60133389423961568180874434594, −3.24884191626243486637931941464, −2.25550588426683540942422150082, −1.60847664258435852494871434923, −0.74067720772506239042240082538,
0.74067720772506239042240082538, 1.60847664258435852494871434923, 2.25550588426683540942422150082, 3.24884191626243486637931941464, 3.60133389423961568180874434594, 4.31690762898032664126534932318, 4.59037596784972644267108292819, 5.00183036684168281129084284042, 5.67339607138908642684889275430, 6.17938178595195748928406873146, 6.76630124393435907765378699325, 7.32469371722216245826150993062, 7.64606590933620344425663891374, 7.968590726590330501123767305785, 8.455818670319071853073892193518
