| L(s) = 1 | − 2·2-s + 3-s + 2·4-s − 2·6-s + 9-s + 3·11-s + 2·12-s − 2·13-s − 4·16-s − 2·18-s − 6·22-s − 2·23-s + 6·25-s + 4·26-s + 27-s + 8·32-s + 3·33-s + 2·36-s − 4·37-s − 2·39-s + 6·44-s + 4·46-s − 4·47-s − 4·48-s − 10·49-s − 12·50-s − 4·52-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s − 0.816·6-s + 1/3·9-s + 0.904·11-s + 0.577·12-s − 0.554·13-s − 16-s − 0.471·18-s − 1.27·22-s − 0.417·23-s + 6/5·25-s + 0.784·26-s + 0.192·27-s + 1.41·32-s + 0.522·33-s + 1/3·36-s − 0.657·37-s − 0.320·39-s + 0.904·44-s + 0.589·46-s − 0.583·47-s − 0.577·48-s − 1.42·49-s − 1.69·50-s − 0.554·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4752 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4752 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5241693756\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5241693756\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | $C_1$ | \( 1 - T \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−12.08408942282026165421113232095, −11.67246952894085626116750430085, −10.87691394832405013001401457429, −10.38936697909055843929130528429, −9.798185870861037407277263459933, −9.202343424123060901986503459506, −8.875144739305145844251978393298, −8.143283213264253094923506804290, −7.62843476028998637591464465653, −6.88163267941314471177005968060, −6.38307042519464451904685536162, −5.06221262167796761274280639428, −4.21718214290935767268143237185, −3.00611925464994524037182969368, −1.66305183534187909223413734470,
1.66305183534187909223413734470, 3.00611925464994524037182969368, 4.21718214290935767268143237185, 5.06221262167796761274280639428, 6.38307042519464451904685536162, 6.88163267941314471177005968060, 7.62843476028998637591464465653, 8.143283213264253094923506804290, 8.875144739305145844251978393298, 9.202343424123060901986503459506, 9.798185870861037407277263459933, 10.38936697909055843929130528429, 10.87691394832405013001401457429, 11.67246952894085626116750430085, 12.08408942282026165421113232095
