| L(s) = 1 | + 3-s + 2·5-s + 9-s + 2·15-s + 4·19-s + 3·25-s + 27-s + 4·43-s + 2·45-s + 2·49-s + 4·57-s + 4·67-s − 12·71-s + 4·73-s + 3·75-s + 81-s + 8·95-s + 16·97-s + 12·101-s + 14·121-s + 4·125-s + 127-s + 4·129-s + 131-s + 2·135-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1/3·9-s + 0.516·15-s + 0.917·19-s + 3/5·25-s + 0.192·27-s + 0.609·43-s + 0.298·45-s + 2/7·49-s + 0.529·57-s + 0.488·67-s − 1.42·71-s + 0.468·73-s + 0.346·75-s + 1/9·81-s + 0.820·95-s + 1.62·97-s + 1.19·101-s + 1.27·121-s + 0.357·125-s + 0.0887·127-s + 0.352·129-s + 0.0873·131-s + 0.172·135-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.546408887\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.546408887\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−9.152090260759482202656303008839, −8.813692459022818092942011437981, −8.350834389912541160170678327096, −7.62723453307151457409205938435, −7.39675758910533928165430736420, −6.80102168565646006174802846887, −6.14633099581892296818931746419, −5.83383772330482403984210898104, −5.11212807204635192120530029111, −4.71942095328565267263165090349, −3.90973085887076474368049757243, −3.31220309478760751953615386764, −2.64551201966495068137482139079, −2.00738971701237820231715588561, −1.10783399282395350103705696425,
1.10783399282395350103705696425, 2.00738971701237820231715588561, 2.64551201966495068137482139079, 3.31220309478760751953615386764, 3.90973085887076474368049757243, 4.71942095328565267263165090349, 5.11212807204635192120530029111, 5.83383772330482403984210898104, 6.14633099581892296818931746419, 6.80102168565646006174802846887, 7.39675758910533928165430736420, 7.62723453307151457409205938435, 8.350834389912541160170678327096, 8.813692459022818092942011437981, 9.152090260759482202656303008839
