| L(s) = 1 | + 2·5-s + 9-s − 4·13-s + 4·17-s + 3·25-s − 20·29-s + 12·37-s − 12·41-s + 2·45-s + 49-s + 12·53-s − 20·61-s − 8·65-s − 28·73-s + 81-s + 8·85-s + 20·89-s + 20·97-s − 20·101-s + 28·109-s − 12·113-s − 4·117-s − 6·121-s + 4·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1/3·9-s − 1.10·13-s + 0.970·17-s + 3/5·25-s − 3.71·29-s + 1.97·37-s − 1.87·41-s + 0.298·45-s + 1/7·49-s + 1.64·53-s − 2.56·61-s − 0.992·65-s − 3.27·73-s + 1/9·81-s + 0.867·85-s + 2.11·89-s + 2.03·97-s − 1.99·101-s + 2.68·109-s − 1.12·113-s − 0.369·117-s − 0.545·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1411200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1411200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 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
−7.64997085218487397133961717810, −7.18329359631779929148111937833, −7.18237449226250170264192150448, −6.14158704727116024821617582822, −6.08759167521302088072182663134, −5.44060214897147557533069905801, −5.29363186716341175587700812987, −4.53435327548118041491444384735, −4.23760798271817481271572497367, −3.39164919683671383154767565220, −3.15914258096067914389137866388, −2.24315879706310572434536107269, −1.93913660460555289279452591140, −1.22215952731265434936026969399, 0,
1.22215952731265434936026969399, 1.93913660460555289279452591140, 2.24315879706310572434536107269, 3.15914258096067914389137866388, 3.39164919683671383154767565220, 4.23760798271817481271572497367, 4.53435327548118041491444384735, 5.29363186716341175587700812987, 5.44060214897147557533069905801, 6.08759167521302088072182663134, 6.14158704727116024821617582822, 7.18237449226250170264192150448, 7.18329359631779929148111937833, 7.64997085218487397133961717810
