L(s) = 1 | + 2-s − 3-s − 4-s − 4·5-s − 6-s − 3·8-s + 9-s − 4·10-s + 12-s + 4·15-s − 16-s + 18-s + 4·20-s + 16·23-s + 3·24-s + 2·25-s − 27-s − 12·29-s + 4·30-s + 5·32-s − 36-s + 12·40-s − 4·45-s + 16·46-s + 16·47-s + 48-s + 2·49-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 1.78·5-s − 0.408·6-s − 1.06·8-s + 1/3·9-s − 1.26·10-s + 0.288·12-s + 1.03·15-s − 1/4·16-s + 0.235·18-s + 0.894·20-s + 3.33·23-s + 0.612·24-s + 2/5·25-s − 0.192·27-s − 2.22·29-s + 0.730·30-s + 0.883·32-s − 1/6·36-s + 1.89·40-s − 0.596·45-s + 2.35·46-s + 2.33·47-s + 0.144·48-s + 2/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209088 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209088 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 3 | $C_1$ | \( 1 + T \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−8.943562213091257213134062166584, −8.495584631629313203905977418995, −7.46995816184461358823952634116, −7.42275429983940735099135130430, −7.18896686257792985441250548424, −6.25377810228368305041472746290, −5.68721248179579351211919147224, −5.31622482563829752699303340190, −4.72191662284056988692815425652, −4.20407903283329927801590362766, −3.81064897716730687178513490998, −3.31425171933473999322603981332, −2.57961275990012009877837343882, −1.06341230628835129193123862023, 0,
1.06341230628835129193123862023, 2.57961275990012009877837343882, 3.31425171933473999322603981332, 3.81064897716730687178513490998, 4.20407903283329927801590362766, 4.72191662284056988692815425652, 5.31622482563829752699303340190, 5.68721248179579351211919147224, 6.25377810228368305041472746290, 7.18896686257792985441250548424, 7.42275429983940735099135130430, 7.46995816184461358823952634116, 8.495584631629313203905977418995, 8.943562213091257213134062166584