| L(s) = 1 | + 5-s + 5·7-s + 11-s − 8·17-s + 4·19-s + 4·23-s + 5·25-s + 10·29-s − 7·31-s + 5·35-s − 8·37-s − 8·41-s + 20·43-s + 6·47-s + 18·49-s − 53-s + 55-s + 9·59-s + 2·61-s − 2·67-s − 12·71-s − 2·73-s + 5·77-s + 9·79-s + 6·83-s − 8·85-s − 6·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.88·7-s + 0.301·11-s − 1.94·17-s + 0.917·19-s + 0.834·23-s + 25-s + 1.85·29-s − 1.25·31-s + 0.845·35-s − 1.31·37-s − 1.24·41-s + 3.04·43-s + 0.875·47-s + 18/7·49-s − 0.137·53-s + 0.134·55-s + 1.17·59-s + 0.256·61-s − 0.244·67-s − 1.42·71-s − 0.234·73-s + 0.569·77-s + 1.01·79-s + 0.658·83-s − 0.867·85-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.031044293\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.031044293\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + T - 52 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 9 T + 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
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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
−9.068327762958757799580664202907, −8.994425015705436011581647918823, −8.495083011601441179985838463628, −8.478657507005296287502795254695, −7.72739416295976960646275621705, −7.35973366164228440268427273115, −6.90347428186296502232815215224, −6.89784169544246728066116567569, −6.09680596053847495464677489411, −5.70365545540187800779698117644, −5.27721344113607903258317983061, −4.88082321367443428607565594741, −4.48542001795131037799890984259, −4.28485181123165383435234236296, −3.52048864702580338578914134711, −2.96558661939030995586903722190, −2.22669533288683847372105892941, −2.10157404764635434982434892135, −1.27287052238439396897820266746, −0.803913703587507309127474327998,
0.803913703587507309127474327998, 1.27287052238439396897820266746, 2.10157404764635434982434892135, 2.22669533288683847372105892941, 2.96558661939030995586903722190, 3.52048864702580338578914134711, 4.28485181123165383435234236296, 4.48542001795131037799890984259, 4.88082321367443428607565594741, 5.27721344113607903258317983061, 5.70365545540187800779698117644, 6.09680596053847495464677489411, 6.89784169544246728066116567569, 6.90347428186296502232815215224, 7.35973366164228440268427273115, 7.72739416295976960646275621705, 8.478657507005296287502795254695, 8.495083011601441179985838463628, 8.994425015705436011581647918823, 9.068327762958757799580664202907
