L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 4·9-s + 2·11-s − 14-s + 16-s − 4·18-s + 2·22-s − 10·23-s − 25-s − 28-s + 6·29-s + 32-s − 4·36-s + 16·37-s + 2·43-s + 2·44-s − 10·46-s + 49-s − 50-s − 14·53-s − 56-s + 6·58-s + 4·63-s + 64-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 4/3·9-s + 0.603·11-s − 0.267·14-s + 1/4·16-s − 0.942·18-s + 0.426·22-s − 2.08·23-s − 1/5·25-s − 0.188·28-s + 1.11·29-s + 0.176·32-s − 2/3·36-s + 2.63·37-s + 0.304·43-s + 0.301·44-s − 1.47·46-s + 1/7·49-s − 0.141·50-s − 1.92·53-s − 0.133·56-s + 0.787·58-s + 0.503·63-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1097600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1097600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_1$ | \( 1 + T \) |
good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
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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
−7.83935170193698139529110089515, −7.54632513625886851614632570414, −6.74305263552018588002663852422, −6.29751439306355490727636887141, −6.13288656637766884974289916676, −5.78715526296965382424297172185, −5.18715695693041182856290290950, −4.60533160312609255643012639607, −4.12297941176406821908955606864, −3.79249297053908164531512164517, −2.95930139589410681750863568790, −2.76020085149736181771668761713, −2.07784112350186327284402004717, −1.19399876736201647235434352923, 0,
1.19399876736201647235434352923, 2.07784112350186327284402004717, 2.76020085149736181771668761713, 2.95930139589410681750863568790, 3.79249297053908164531512164517, 4.12297941176406821908955606864, 4.60533160312609255643012639607, 5.18715695693041182856290290950, 5.78715526296965382424297172185, 6.13288656637766884974289916676, 6.29751439306355490727636887141, 6.74305263552018588002663852422, 7.54632513625886851614632570414, 7.83935170193698139529110089515