| L(s) = 1 | + 2·3-s + 9-s + 4·11-s − 4·23-s − 5·25-s − 4·27-s + 2·31-s + 8·33-s − 6·37-s − 10·47-s − 4·49-s − 8·53-s + 16·67-s − 8·69-s − 8·71-s − 10·75-s − 11·81-s − 2·89-s + 4·93-s − 22·97-s + 4·99-s − 8·103-s − 12·111-s + 20·113-s + 5·121-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s + 1.20·11-s − 0.834·23-s − 25-s − 0.769·27-s + 0.359·31-s + 1.39·33-s − 0.986·37-s − 1.45·47-s − 4/7·49-s − 1.09·53-s + 1.95·67-s − 0.963·69-s − 0.949·71-s − 1.15·75-s − 1.22·81-s − 0.211·89-s + 0.414·93-s − 2.23·97-s + 0.402·99-s − 0.788·103-s − 1.13·111-s + 1.88·113-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + p T^{2} \) |
| 11 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
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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
−7.86735145256006128225680914884, −7.12677891119421151760402250886, −6.87192318854660231304818254218, −6.39905752530911314023527935340, −5.92898007255709140054382921533, −5.52146253379322057238372662992, −4.89089994467430196771545977685, −4.31817677098915716449378029039, −3.96392870128622868080360965231, −3.39484621967365686957833988089, −3.16857164585558976996154852725, −2.34262874507074568590808082837, −1.86687205836808265029991573209, −1.33036298878947812916571172751, 0,
1.33036298878947812916571172751, 1.86687205836808265029991573209, 2.34262874507074568590808082837, 3.16857164585558976996154852725, 3.39484621967365686957833988089, 3.96392870128622868080360965231, 4.31817677098915716449378029039, 4.89089994467430196771545977685, 5.52146253379322057238372662992, 5.92898007255709140054382921533, 6.39905752530911314023527935340, 6.87192318854660231304818254218, 7.12677891119421151760402250886, 7.86735145256006128225680914884
