| L(s) = 1 | − 2·5-s + 9-s + 4·13-s − 12·17-s + 3·25-s + 4·29-s − 12·37-s − 12·41-s − 2·45-s + 2·49-s + 20·53-s + 20·61-s − 8·65-s + 20·73-s + 81-s + 24·85-s − 12·89-s + 36·97-s − 12·101-s − 12·109-s − 12·113-s + 4·117-s − 22·121-s − 4·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1/3·9-s + 1.10·13-s − 2.91·17-s + 3/5·25-s + 0.742·29-s − 1.97·37-s − 1.87·41-s − 0.298·45-s + 2/7·49-s + 2.74·53-s + 2.56·61-s − 0.992·65-s + 2.34·73-s + 1/9·81-s + 2.60·85-s − 1.27·89-s + 3.65·97-s − 1.19·101-s − 1.14·109-s − 1.12·113-s + 0.369·117-s − 2·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
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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
−8.111657163103838072854194916622, −7.45134778749952815389203872692, −6.79900118215484039132172322730, −6.70226451252379052547279711019, −6.57738838391963626904005050389, −5.57736353128468961141160211811, −5.18679626504361739623813980970, −4.70648708629361190231505353737, −4.04830830196212021008842338792, −3.87106400853132584583260410760, −3.35245252789423871046146180991, −2.39515917409923388922495134035, −2.06437956968094832990073177575, −1.03399709961917238474663824042, 0,
1.03399709961917238474663824042, 2.06437956968094832990073177575, 2.39515917409923388922495134035, 3.35245252789423871046146180991, 3.87106400853132584583260410760, 4.04830830196212021008842338792, 4.70648708629361190231505353737, 5.18679626504361739623813980970, 5.57736353128468961141160211811, 6.57738838391963626904005050389, 6.70226451252379052547279711019, 6.79900118215484039132172322730, 7.45134778749952815389203872692, 8.111657163103838072854194916622
