| L(s) = 1 | + 2·2-s − 3-s + 2·4-s + 4·5-s − 2·6-s − 2·9-s + 8·10-s − 2·12-s − 4·15-s − 4·16-s − 4·18-s − 5·19-s + 8·20-s − 8·23-s + 11·25-s + 5·27-s − 8·30-s − 8·32-s − 4·36-s − 10·38-s + 8·43-s − 8·45-s − 16·46-s + 4·47-s + 4·48-s + 5·49-s + 22·50-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s + 1.78·5-s − 0.816·6-s − 2/3·9-s + 2.52·10-s − 0.577·12-s − 1.03·15-s − 16-s − 0.942·18-s − 1.14·19-s + 1.78·20-s − 1.66·23-s + 11/5·25-s + 0.962·27-s − 1.46·30-s − 1.41·32-s − 2/3·36-s − 1.62·38-s + 1.21·43-s − 1.19·45-s − 2.35·46-s + 0.583·47-s + 0.577·48-s + 5/7·49-s + 3.11·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.071314781\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.071314781\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 73 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
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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
−11.17890422995367215770445868093, −10.78709634204626383703352437221, −10.22212301104304794804317557426, −9.656975297597193557078061547607, −8.982400780552325097117784054768, −8.572472937030225465369476645071, −7.59675340173858040698590874132, −6.59997372874912544363684164399, −6.20829491400098211939655341817, −5.88445921386218300839462084180, −5.32369250416837801662678140356, −4.67311448273952802484963300668, −3.86148951367411079751835480875, −2.71571062617939245109812166064, −2.08110417889891008752239981228,
2.08110417889891008752239981228, 2.71571062617939245109812166064, 3.86148951367411079751835480875, 4.67311448273952802484963300668, 5.32369250416837801662678140356, 5.88445921386218300839462084180, 6.20829491400098211939655341817, 6.59997372874912544363684164399, 7.59675340173858040698590874132, 8.572472937030225465369476645071, 8.982400780552325097117784054768, 9.656975297597193557078061547607, 10.22212301104304794804317557426, 10.78709634204626383703352437221, 11.17890422995367215770445868093
