| L(s) = 1 | + 2·5-s − 8·7-s − 6·9-s + 4·11-s + 8·19-s + 3·25-s − 16·35-s + 12·37-s − 16·43-s − 12·45-s + 34·49-s + 12·53-s + 8·55-s + 48·63-s − 32·77-s + 27·81-s − 32·83-s − 12·89-s + 16·95-s − 28·97-s − 24·99-s + 36·113-s + 5·121-s + 4·125-s + 127-s + 131-s − 64·133-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 3.02·7-s − 2·9-s + 1.20·11-s + 1.83·19-s + 3/5·25-s − 2.70·35-s + 1.97·37-s − 2.43·43-s − 1.78·45-s + 34/7·49-s + 1.64·53-s + 1.07·55-s + 6.04·63-s − 3.64·77-s + 3·81-s − 3.51·83-s − 1.27·89-s + 1.64·95-s − 2.84·97-s − 2.41·99-s + 3.38·113-s + 5/11·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s − 5.54·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−8.563879011661793903559115865259, −8.223522789927760706880481247520, −7.17333521505739308879256846630, −6.97238544391138654785050907556, −6.51302590756949038823070214927, −6.02838569034650388568211971598, −5.70093888866808737514223874840, −5.54153279390545819436346988340, −4.53243901735140236687311130642, −3.66574171865656283286960534597, −3.26180587408034592610487247010, −2.94467874037426815586288045320, −2.44821359744315819514002541053, −1.09598094259623312561400782139, 0,
1.09598094259623312561400782139, 2.44821359744315819514002541053, 2.94467874037426815586288045320, 3.26180587408034592610487247010, 3.66574171865656283286960534597, 4.53243901735140236687311130642, 5.54153279390545819436346988340, 5.70093888866808737514223874840, 6.02838569034650388568211971598, 6.51302590756949038823070214927, 6.97238544391138654785050907556, 7.17333521505739308879256846630, 8.223522789927760706880481247520, 8.563879011661793903559115865259
