| L(s) = 1 | − 2·2-s + 3·4-s − 2·7-s − 4·8-s − 2·11-s + 4·14-s + 5·16-s + 4·22-s − 6·28-s + 2·29-s − 6·32-s + 2·37-s + 2·43-s − 6·44-s + 3·49-s + 2·53-s + 8·56-s − 4·58-s + 7·64-s + 2·67-s − 4·74-s + 4·77-s − 81-s − 4·86-s + 8·88-s − 6·98-s − 4·106-s + ⋯ |
| L(s) = 1 | − 2·2-s + 3·4-s − 2·7-s − 4·8-s − 2·11-s + 4·14-s + 5·16-s + 4·22-s − 6·28-s + 2·29-s − 6·32-s + 2·37-s + 2·43-s − 6·44-s + 3·49-s + 2·53-s + 8·56-s − 4·58-s + 7·64-s + 2·67-s − 4·74-s + 4·77-s − 81-s − 4·86-s + 8·88-s − 6·98-s − 4·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3156081019\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3156081019\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T^{4} \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + T^{4} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + T^{4} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + T^{4} \) |
| 61 | $C_2^2$ | \( 1 + T^{4} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + T^{4} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
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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
−9.200472253571314502173144251965, −8.851910939477824819076561951213, −8.430060305915447396797415858500, −8.116612011059271765159953584554, −7.61365625356703023209743777582, −7.58180308734672494215839796834, −6.88593754623943232424268765797, −6.72733974255722542258131088129, −6.33969862883068448902165895606, −5.92352991805528178600974255533, −5.47039880041512502445040689723, −5.31736595214568411602469463324, −4.26107348719692623995583089648, −3.92505069827376458488398186427, −3.11353231397606695342027140404, −2.83060586188879965332347039221, −2.51598418070910656439914318170, −2.28995185911227562548074960459, −1.04777783004992994368628790204, −0.58019927509636180084218594227,
0.58019927509636180084218594227, 1.04777783004992994368628790204, 2.28995185911227562548074960459, 2.51598418070910656439914318170, 2.83060586188879965332347039221, 3.11353231397606695342027140404, 3.92505069827376458488398186427, 4.26107348719692623995583089648, 5.31736595214568411602469463324, 5.47039880041512502445040689723, 5.92352991805528178600974255533, 6.33969862883068448902165895606, 6.72733974255722542258131088129, 6.88593754623943232424268765797, 7.58180308734672494215839796834, 7.61365625356703023209743777582, 8.116612011059271765159953584554, 8.430060305915447396797415858500, 8.851910939477824819076561951213, 9.200472253571314502173144251965
