| L(s) = 1 | + 2·5-s + 2·7-s − 2·11-s + 2·17-s + 25-s + 4·35-s − 2·43-s − 2·47-s + 49-s − 4·55-s − 2·61-s + 2·73-s − 4·77-s − 81-s + 4·85-s + 4·119-s + 121-s − 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 2·5-s + 2·7-s − 2·11-s + 2·17-s + 25-s + 4·35-s − 2·43-s − 2·47-s + 49-s − 4·55-s − 2·61-s + 2·73-s − 4·77-s − 81-s + 4·85-s + 4·119-s + 121-s − 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2085136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2085136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.891072969\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.891072969\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 19 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + T^{4} \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + T^{4} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_2^2$ | \( 1 + T^{4} \) |
| 41 | $C_2^2$ | \( 1 + T^{4} \) |
| 43 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2^2$ | \( 1 + T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 71 | $C_2^2$ | \( 1 + T^{4} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−9.982688477019019676449701848876, −9.541781335649456715839484153441, −9.383922560218654592964385778835, −8.478877344675692620429186750915, −8.327991113120855842680468721897, −7.84055444481413429844057499486, −7.78092285616281715056948987331, −7.26487135219802101101900613453, −6.39945860036386387640003245147, −6.27654646058835998443996530083, −5.54853898334982717113260319787, −5.31016747309092775443413896561, −5.12110165769165580279851583405, −4.82467906905351532808468452381, −4.05794158105469510604608631770, −3.10830175164390682458008859562, −2.96572444043556335435036459743, −2.03785135530993626435906984304, −1.80070858114653563976834752223, −1.36050268332993543305465509124,
1.36050268332993543305465509124, 1.80070858114653563976834752223, 2.03785135530993626435906984304, 2.96572444043556335435036459743, 3.10830175164390682458008859562, 4.05794158105469510604608631770, 4.82467906905351532808468452381, 5.12110165769165580279851583405, 5.31016747309092775443413896561, 5.54853898334982717113260319787, 6.27654646058835998443996530083, 6.39945860036386387640003245147, 7.26487135219802101101900613453, 7.78092285616281715056948987331, 7.84055444481413429844057499486, 8.327991113120855842680468721897, 8.478877344675692620429186750915, 9.383922560218654592964385778835, 9.541781335649456715839484153441, 9.982688477019019676449701848876
