| L(s) = 1 | + 2·2-s + 2·4-s + 2·5-s + 4·8-s + 4·10-s + 2·11-s − 8·13-s + 8·16-s + 10·17-s − 2·19-s + 4·20-s + 4·22-s + 6·23-s + 3·25-s − 16·26-s − 2·29-s − 6·31-s + 8·32-s + 20·34-s + 4·37-s − 4·38-s + 8·40-s + 2·41-s − 4·43-s + 4·44-s + 12·46-s + 4·47-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.894·5-s + 1.41·8-s + 1.26·10-s + 0.603·11-s − 2.21·13-s + 2·16-s + 2.42·17-s − 0.458·19-s + 0.894·20-s + 0.852·22-s + 1.25·23-s + 3/5·25-s − 3.13·26-s − 0.371·29-s − 1.07·31-s + 1.41·32-s + 3.42·34-s + 0.657·37-s − 0.648·38-s + 1.26·40-s + 0.312·41-s − 0.609·43-s + 0.603·44-s + 1.76·46-s + 0.583·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.141733387\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.141733387\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 20 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 8 T + 3 p T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 10 T + 56 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 52 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 32 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 59 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 2 T + 80 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 63 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 10 T + 116 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 95 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 2 T + 116 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 87 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 6 T + 59 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 40 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 16 T + 210 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
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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
−9.387169424041738946575738682102, −9.070815285831849735504544354741, −8.329314854553710522898125541166, −7.901649637049607527798861593735, −7.53449583874451770876862790834, −7.18116904729396565300064076755, −7.02791875543468575873253069303, −6.41064938264516690572384493412, −5.76823206927625251967800913820, −5.65869401239177399362902694186, −5.22465764297220065526974682434, −4.94303185529883518136152078696, −4.40789050424413627140145966940, −4.19671804362390176630651257814, −3.27809948990375726129320916217, −3.26233120230207922111087899340, −2.62354844400724227469842370311, −1.94886034724864169709111293311, −1.56034307809233887657442766841, −0.801804638470364563742738054091,
0.801804638470364563742738054091, 1.56034307809233887657442766841, 1.94886034724864169709111293311, 2.62354844400724227469842370311, 3.26233120230207922111087899340, 3.27809948990375726129320916217, 4.19671804362390176630651257814, 4.40789050424413627140145966940, 4.94303185529883518136152078696, 5.22465764297220065526974682434, 5.65869401239177399362902694186, 5.76823206927625251967800913820, 6.41064938264516690572384493412, 7.02791875543468575873253069303, 7.18116904729396565300064076755, 7.53449583874451770876862790834, 7.901649637049607527798861593735, 8.329314854553710522898125541166, 9.070815285831849735504544354741, 9.387169424041738946575738682102
