| L(s) = 1 | − 2-s + 4-s − 5-s + 4·7-s − 8-s + 9-s + 10-s + 6·13-s − 4·14-s + 16-s − 18-s − 20-s − 4·25-s − 6·26-s + 4·28-s + 8·29-s − 32-s − 4·35-s + 36-s − 10·37-s + 40-s − 45-s + 4·47-s + 7·49-s + 4·50-s + 6·52-s − 4·56-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.66·13-s − 1.06·14-s + 1/4·16-s − 0.235·18-s − 0.223·20-s − 4/5·25-s − 1.17·26-s + 0.755·28-s + 1.48·29-s − 0.176·32-s − 0.676·35-s + 1/6·36-s − 1.64·37-s + 0.158·40-s − 0.149·45-s + 0.583·47-s + 49-s + 0.565·50-s + 0.832·52-s − 0.534·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 540800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.783577051\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.783577051\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 5 | $C_2$ | \( 1 + T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 39 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 47 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
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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
−8.441478766003148556623796299623, −8.144095561645493793928142696150, −7.79553307742508526649912506501, −7.15207845955938326404734101537, −6.77144560653655953790652734450, −6.33155021821359964214935134347, −5.66369001367489491921436286434, −5.24767560053836964503588703269, −4.72221612142743047282768708968, −3.97326277707734270004818295514, −3.79580057738805113963063133049, −2.95435702019764121095310342586, −2.10228754368625768564730352600, −1.53550791444121486048744677720, −0.856906510162455468474606832055,
0.856906510162455468474606832055, 1.53550791444121486048744677720, 2.10228754368625768564730352600, 2.95435702019764121095310342586, 3.79580057738805113963063133049, 3.97326277707734270004818295514, 4.72221612142743047282768708968, 5.24767560053836964503588703269, 5.66369001367489491921436286434, 6.33155021821359964214935134347, 6.77144560653655953790652734450, 7.15207845955938326404734101537, 7.79553307742508526649912506501, 8.144095561645493793928142696150, 8.441478766003148556623796299623
