| L(s) = 1 | + 3-s + 5-s − 2·9-s + 15-s − 5·19-s − 6·23-s + 25-s − 5·27-s − 6·29-s + 4·43-s − 2·45-s − 12·47-s + 5·49-s + 6·53-s − 5·57-s − 5·67-s − 6·69-s − 6·71-s + 4·73-s + 75-s + 81-s − 6·87-s − 5·95-s − 20·97-s − 24·101-s − 6·115-s + 5·121-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 2/3·9-s + 0.258·15-s − 1.14·19-s − 1.25·23-s + 1/5·25-s − 0.962·27-s − 1.11·29-s + 0.609·43-s − 0.298·45-s − 1.75·47-s + 5/7·49-s + 0.824·53-s − 0.662·57-s − 0.610·67-s − 0.722·69-s − 0.712·71-s + 0.468·73-s + 0.115·75-s + 1/9·81-s − 0.643·87-s − 0.512·95-s − 2.03·97-s − 2.38·101-s − 0.559·115-s + 5/11·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 5 | $C_1$ | \( 1 - T \) |
| good | 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 37 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
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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
−8.588643869456477815775703266121, −8.177103512204866898480973830433, −7.890500500715917472230954666010, −7.23364004848274182350526794123, −6.72357923653185277629621165847, −6.20622892935818810426279816609, −5.68541060496024950379246444775, −5.42934568072141405308354593734, −4.55892161728402244260025936292, −4.05092172773156110742028983426, −3.53505714272045056127314576009, −2.76278360271833845392651601641, −2.24547352231884341786124802358, −1.59613551852467409546683372355, 0,
1.59613551852467409546683372355, 2.24547352231884341786124802358, 2.76278360271833845392651601641, 3.53505714272045056127314576009, 4.05092172773156110742028983426, 4.55892161728402244260025936292, 5.42934568072141405308354593734, 5.68541060496024950379246444775, 6.20622892935818810426279816609, 6.72357923653185277629621165847, 7.23364004848274182350526794123, 7.890500500715917472230954666010, 8.177103512204866898480973830433, 8.588643869456477815775703266121
