| L(s) = 1 | + 4-s − 9-s − 4·11-s + 2·13-s − 3·16-s − 2·17-s − 2·25-s − 6·29-s − 36-s − 6·37-s − 8·43-s − 4·44-s − 4·47-s + 49-s + 2·52-s + 6·53-s − 8·59-s − 7·64-s − 2·68-s − 8·81-s + 4·89-s + 6·97-s + 4·99-s − 2·100-s + 20·107-s − 10·113-s − 6·116-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 1/3·9-s − 1.20·11-s + 0.554·13-s − 3/4·16-s − 0.485·17-s − 2/5·25-s − 1.11·29-s − 1/6·36-s − 0.986·37-s − 1.21·43-s − 0.603·44-s − 0.583·47-s + 1/7·49-s + 0.277·52-s + 0.824·53-s − 1.04·59-s − 7/8·64-s − 0.242·68-s − 8/9·81-s + 0.423·89-s + 0.609·97-s + 0.402·99-s − 1/5·100-s + 1.93·107-s − 0.940·113-s − 0.557·116-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 137641 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137641 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 53 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 33 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 85 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 87 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - T + p T^{2} ) \) |
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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
−8.965189545668973371850831140050, −8.675697572992931760415878019149, −8.133644721434451692190571994972, −7.60013928785060028593997498047, −7.18729262571201574330029560168, −6.62224367875363672581460941647, −6.13324145080416111769745910300, −5.55899470277431583414593672632, −5.07051024096508536713220740245, −4.47130801306741884223475447311, −3.68520704670013565965267999575, −3.08688213013088840571554219942, −2.34036015708209897566945451388, −1.72070972811502253177948294750, 0,
1.72070972811502253177948294750, 2.34036015708209897566945451388, 3.08688213013088840571554219942, 3.68520704670013565965267999575, 4.47130801306741884223475447311, 5.07051024096508536713220740245, 5.55899470277431583414593672632, 6.13324145080416111769745910300, 6.62224367875363672581460941647, 7.18729262571201574330029560168, 7.60013928785060028593997498047, 8.133644721434451692190571994972, 8.675697572992931760415878019149, 8.965189545668973371850831140050
