| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s − 2·9-s + 12-s + 16-s + 2·18-s + 2·19-s + 6·23-s − 24-s − 10·25-s − 5·27-s + 18·29-s − 32-s − 2·36-s − 2·38-s + 16·43-s − 6·46-s + 48-s − 13·49-s + 10·50-s − 6·53-s + 5·54-s + 2·57-s − 18·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s − 2/3·9-s + 0.288·12-s + 1/4·16-s + 0.471·18-s + 0.458·19-s + 1.25·23-s − 0.204·24-s − 2·25-s − 0.962·27-s + 3.34·29-s − 0.176·32-s − 1/3·36-s − 0.324·38-s + 2.43·43-s − 0.884·46-s + 0.144·48-s − 1.85·49-s + 1.41·50-s − 0.824·53-s + 0.680·54-s + 0.264·57-s − 2.36·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 415872 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 415872 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.459279525\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.459279525\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_2$ | \( 1 - T + p T^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−8.684413720393242419277779815238, −8.190910734073893474198823012733, −7.73591843742988285704805588989, −7.58003549586688296986370886186, −6.73344818810584336562000567981, −6.49666840322324443514516121077, −5.81027585652647529822118508915, −5.53558881857441470981084044722, −4.63536113703220970332946568314, −4.35838689667544813938052375431, −3.38077129601560614051704553932, −2.99432139105097413830284496966, −2.51502825919439612349923033615, −1.68287474868974763875079964203, −0.72725765709121216894521989048,
0.72725765709121216894521989048, 1.68287474868974763875079964203, 2.51502825919439612349923033615, 2.99432139105097413830284496966, 3.38077129601560614051704553932, 4.35838689667544813938052375431, 4.63536113703220970332946568314, 5.53558881857441470981084044722, 5.81027585652647529822118508915, 6.49666840322324443514516121077, 6.73344818810584336562000567981, 7.58003549586688296986370886186, 7.73591843742988285704805588989, 8.190910734073893474198823012733, 8.684413720393242419277779815238
