L(s) = 1 | + 2-s + 4-s + 8-s − 5·9-s + 16-s − 5·18-s + 6·25-s − 12·29-s + 32-s − 5·36-s − 14·37-s − 7·49-s + 6·50-s − 4·53-s − 12·58-s + 64-s − 5·72-s − 14·74-s + 16·81-s − 7·98-s + 6·100-s − 4·106-s − 4·109-s + 10·113-s − 12·116-s − 21·121-s + 127-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 5/3·9-s + 1/4·16-s − 1.17·18-s + 6/5·25-s − 2.22·29-s + 0.176·32-s − 5/6·36-s − 2.30·37-s − 49-s + 0.848·50-s − 0.549·53-s − 1.57·58-s + 1/8·64-s − 0.589·72-s − 1.62·74-s + 16/9·81-s − 0.707·98-s + 3/5·100-s − 0.388·106-s − 0.383·109-s + 0.940·113-s − 1.11·116-s − 1.90·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 264992 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264992 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 95 T^{2} + p^{2} T^{4} \) |
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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
−8.685004820357231702506420926408, −8.248439315301770823500061130309, −7.63337800719004659346001520913, −7.29364185989773286492114805717, −6.52782796668404364019147525307, −6.35745039777708092394304246751, −5.55701452212892334338525238478, −5.31122646603514580888005499368, −4.92952659196260751010674746014, −4.04787440914434876138270957457, −3.46526394563144742666387467850, −3.07585320913390325960408879373, −2.35878535938057985397491679895, −1.58472569577804288902591271666, 0,
1.58472569577804288902591271666, 2.35878535938057985397491679895, 3.07585320913390325960408879373, 3.46526394563144742666387467850, 4.04787440914434876138270957457, 4.92952659196260751010674746014, 5.31122646603514580888005499368, 5.55701452212892334338525238478, 6.35745039777708092394304246751, 6.52782796668404364019147525307, 7.29364185989773286492114805717, 7.63337800719004659346001520913, 8.248439315301770823500061130309, 8.685004820357231702506420926408