| L(s) = 1 | + 2·2-s + 2·3-s + 4-s − 2·5-s + 4·6-s − 9-s − 4·10-s + 2·12-s + 2·13-s − 4·15-s + 16-s + 4·17-s − 2·18-s − 12·19-s − 2·20-s − 4·23-s − 7·25-s + 4·26-s − 6·27-s − 2·29-s − 8·30-s + 6·31-s − 2·32-s + 8·34-s − 36-s − 8·37-s − 24·38-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 1/2·4-s − 0.894·5-s + 1.63·6-s − 1/3·9-s − 1.26·10-s + 0.577·12-s + 0.554·13-s − 1.03·15-s + 1/4·16-s + 0.970·17-s − 0.471·18-s − 2.75·19-s − 0.447·20-s − 0.834·23-s − 7/5·25-s + 0.784·26-s − 1.15·27-s − 0.371·29-s − 1.46·30-s + 1.07·31-s − 0.353·32-s + 1.37·34-s − 1/6·36-s − 1.31·37-s − 3.89·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12313081 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12313081 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8742023612\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8742023612\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 11 | | \( 1 \) |
| 29 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - p T + 3 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 19 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 21 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 26 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 10 T + 109 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2 T + 77 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 35 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 71 | $C_4$ | \( 1 + 12 T + 170 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 157 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 138 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 8 T + 122 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 138 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−8.501004066979581415725685656766, −8.413152084206653574407718801436, −8.062113844637639361137633564058, −7.891617029235043184798936573683, −7.19824456068185246578110689099, −6.99076623090617137646970973152, −6.24512046106919348745106744356, −6.20576779459087039239907196259, −5.64307208007198149911540016666, −5.34085656324126785384594814360, −4.77337554252691899539708668888, −4.37189241072126895870212737877, −4.05846744986970638443156044361, −3.77669700063362641136543093508, −3.29255895514306354773119552513, −3.19321184265183032661334866547, −2.42098747962863390906768644384, −1.92502704300329934886535612787, −1.59038696780108382933866974932, −0.18645900038238421833756067869,
0.18645900038238421833756067869, 1.59038696780108382933866974932, 1.92502704300329934886535612787, 2.42098747962863390906768644384, 3.19321184265183032661334866547, 3.29255895514306354773119552513, 3.77669700063362641136543093508, 4.05846744986970638443156044361, 4.37189241072126895870212737877, 4.77337554252691899539708668888, 5.34085656324126785384594814360, 5.64307208007198149911540016666, 6.20576779459087039239907196259, 6.24512046106919348745106744356, 6.99076623090617137646970973152, 7.19824456068185246578110689099, 7.891617029235043184798936573683, 8.062113844637639361137633564058, 8.413152084206653574407718801436, 8.501004066979581415725685656766
