| L(s) = 1 | − 4·5-s − 9-s + 8·11-s + 12·19-s + 11·25-s − 12·29-s + 4·31-s + 16·41-s + 4·45-s − 49-s − 32·55-s + 8·59-s + 28·61-s + 81-s − 16·89-s − 48·95-s − 8·99-s + 32·101-s + 28·109-s + 26·121-s − 24·125-s + 127-s + 131-s + 137-s + 139-s + 48·145-s + 149-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 1/3·9-s + 2.41·11-s + 2.75·19-s + 11/5·25-s − 2.22·29-s + 0.718·31-s + 2.49·41-s + 0.596·45-s − 1/7·49-s − 4.31·55-s + 1.04·59-s + 3.58·61-s + 1/9·81-s − 1.69·89-s − 4.92·95-s − 0.804·99-s + 3.18·101-s + 2.68·109-s + 2.36·121-s − 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3.98·145-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2822400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2822400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.323890274\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.323890274\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−9.488183623370718288035170545107, −9.137980944463362249593741490558, −8.616106659845160174059244919632, −8.578180954717967574982679366391, −7.83696356226028951202082917396, −7.47284940877725177831588395027, −7.15798338761632556354860069909, −7.12963352568719928396862848136, −6.23263812359207440330252395833, −6.09743681843731010569910737818, −5.30397512914412908283084271049, −5.15449796315116488801900665728, −4.31160241874633658516906267280, −4.04747557179696972558413138932, −3.62796829751011359702272528238, −3.42031777145297627005271159277, −2.77274814962515678305357825915, −1.94256030229403833502821782546, −0.935188665223478661022326246369, −0.841417224320144161065720466590,
0.841417224320144161065720466590, 0.935188665223478661022326246369, 1.94256030229403833502821782546, 2.77274814962515678305357825915, 3.42031777145297627005271159277, 3.62796829751011359702272528238, 4.04747557179696972558413138932, 4.31160241874633658516906267280, 5.15449796315116488801900665728, 5.30397512914412908283084271049, 6.09743681843731010569910737818, 6.23263812359207440330252395833, 7.12963352568719928396862848136, 7.15798338761632556354860069909, 7.47284940877725177831588395027, 7.83696356226028951202082917396, 8.578180954717967574982679366391, 8.616106659845160174059244919632, 9.137980944463362249593741490558, 9.488183623370718288035170545107
