L(s) = 1 | − 3-s − 2·9-s − 3·11-s + 3·17-s + 6·19-s − 3·25-s + 5·27-s + 3·33-s − 6·41-s + 49-s − 3·51-s − 6·57-s − 3·59-s − 12·67-s − 18·73-s + 3·75-s + 81-s + 9·83-s − 13·89-s + 7·97-s + 6·99-s + 27·107-s + 19·113-s + 5·121-s + 6·123-s + 127-s + 131-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 2/3·9-s − 0.904·11-s + 0.727·17-s + 1.37·19-s − 3/5·25-s + 0.962·27-s + 0.522·33-s − 0.937·41-s + 1/7·49-s − 0.420·51-s − 0.794·57-s − 0.390·59-s − 1.46·67-s − 2.10·73-s + 0.346·75-s + 1/9·81-s + 0.987·83-s − 1.37·89-s + 0.710·97-s + 0.603·99-s + 2.61·107-s + 1.78·113-s + 5/11·121-s + 0.541·123-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 903168 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 903168 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 5 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 21 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 56 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 41 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 63 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−7.79617395585658449749048674131, −7.58786261587115276945767070016, −7.17163860203204208317008760634, −6.58669237598845274008443210105, −5.96741850796570975556193970452, −5.71658687274490405209558062284, −5.38466494213673094678859456962, −4.77928753297053300382547797947, −4.48319472456144754515654431837, −3.48746963948844285830900733940, −3.23226123087442130863517621395, −2.67484893251737962938140937780, −1.87364629480523355719376460556, −1.01439625642395527691688521044, 0,
1.01439625642395527691688521044, 1.87364629480523355719376460556, 2.67484893251737962938140937780, 3.23226123087442130863517621395, 3.48746963948844285830900733940, 4.48319472456144754515654431837, 4.77928753297053300382547797947, 5.38466494213673094678859456962, 5.71658687274490405209558062284, 5.96741850796570975556193970452, 6.58669237598845274008443210105, 7.17163860203204208317008760634, 7.58786261587115276945767070016, 7.79617395585658449749048674131