L(s) = 1 | − 4·4-s − 2·7-s − 5·9-s − 2·11-s + 12·16-s + 14·23-s − 9·25-s + 8·28-s − 4·29-s + 20·36-s − 22·37-s − 8·43-s + 8·44-s − 3·49-s + 4·53-s + 10·63-s − 32·64-s − 2·67-s − 18·71-s + 4·77-s + 16·79-s + 16·81-s − 56·92-s + 10·99-s + 36·100-s + 16·107-s + 8·109-s + ⋯ |
L(s) = 1 | − 2·4-s − 0.755·7-s − 5/3·9-s − 0.603·11-s + 3·16-s + 2.91·23-s − 9/5·25-s + 1.51·28-s − 0.742·29-s + 10/3·36-s − 3.61·37-s − 1.21·43-s + 1.20·44-s − 3/7·49-s + 0.549·53-s + 1.25·63-s − 4·64-s − 0.244·67-s − 2.13·71-s + 0.455·77-s + 1.80·79-s + 16/9·81-s − 5.83·92-s + 1.00·99-s + 18/5·100-s + 1.54·107-s + 0.766·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1002001 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1002001 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 7 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
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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
−7.85612481771897939782236601586, −7.21661224338768342466420756206, −6.96627157214221326686940120925, −6.05777360202583450452013331209, −5.82135849339387357739621546932, −5.31104352782074944000454805478, −4.95237769114385482752321677072, −4.75768041533452306802121891710, −3.70099161938712630953878841304, −3.35775542055244311804083250514, −3.29304594675951575785139026233, −2.34448923217473262650057367432, −1.28655472231449922500586089553, 0, 0,
1.28655472231449922500586089553, 2.34448923217473262650057367432, 3.29304594675951575785139026233, 3.35775542055244311804083250514, 3.70099161938712630953878841304, 4.75768041533452306802121891710, 4.95237769114385482752321677072, 5.31104352782074944000454805478, 5.82135849339387357739621546932, 6.05777360202583450452013331209, 6.96627157214221326686940120925, 7.21661224338768342466420756206, 7.85612481771897939782236601586