L(s) = 1 | + 2-s − 3-s − 4-s − 6-s − 3·8-s + 9-s + 8·11-s + 12-s + 2·13-s − 16-s + 18-s + 8·22-s + 3·24-s − 6·25-s + 2·26-s − 27-s + 5·32-s − 8·33-s − 36-s − 4·37-s − 2·39-s − 8·44-s + 48-s + 2·49-s − 6·50-s − 2·52-s − 54-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s − 1.06·8-s + 1/3·9-s + 2.41·11-s + 0.288·12-s + 0.554·13-s − 1/4·16-s + 0.235·18-s + 1.70·22-s + 0.612·24-s − 6/5·25-s + 0.392·26-s − 0.192·27-s + 0.883·32-s − 1.39·33-s − 1/6·36-s − 0.657·37-s − 0.320·39-s − 1.20·44-s + 0.144·48-s + 2/7·49-s − 0.848·50-s − 0.277·52-s − 0.136·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73008 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73008 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.578274261\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.578274261\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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
−9.952802346640542280811898231944, −9.192740554674130435254875131876, −8.921039436304358900231528874001, −8.484648143275568894036347393857, −7.73604447848719394517643815023, −6.99405970659997290412829823598, −6.54556545042783689480045128663, −6.08826329831047007406167672262, −5.65486780279705070512904066158, −4.94568196736683705255358366288, −4.31391709618999720844014512009, −3.71205447024015551816570926916, −3.55158930907206838697007657967, −2.10208151844778724001409156325, −1.00171028893857761930540835980,
1.00171028893857761930540835980, 2.10208151844778724001409156325, 3.55158930907206838697007657967, 3.71205447024015551816570926916, 4.31391709618999720844014512009, 4.94568196736683705255358366288, 5.65486780279705070512904066158, 6.08826329831047007406167672262, 6.54556545042783689480045128663, 6.99405970659997290412829823598, 7.73604447848719394517643815023, 8.484648143275568894036347393857, 8.921039436304358900231528874001, 9.192740554674130435254875131876, 9.952802346640542280811898231944