| L(s) = 1 | + 2·2-s + 3-s + 3·4-s + 2·6-s + 7-s + 4·8-s − 9-s + 8·11-s + 3·12-s + 13-s + 2·14-s + 5·16-s − 11·17-s − 2·18-s − 2·19-s + 21-s + 16·22-s − 3·23-s + 4·24-s + 2·26-s + 3·28-s + 29-s − 2·31-s + 6·32-s + 8·33-s − 22·34-s − 3·36-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 3/2·4-s + 0.816·6-s + 0.377·7-s + 1.41·8-s − 1/3·9-s + 2.41·11-s + 0.866·12-s + 0.277·13-s + 0.534·14-s + 5/4·16-s − 2.66·17-s − 0.471·18-s − 0.458·19-s + 0.218·21-s + 3.41·22-s − 0.625·23-s + 0.816·24-s + 0.392·26-s + 0.566·28-s + 0.185·29-s − 0.359·31-s + 1.06·32-s + 1.39·33-s − 3.77·34-s − 1/2·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.526327402\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.526327402\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - T + 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - T + 20 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 46 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 14 T + 118 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 5 T + 108 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - T + 114 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - T + 130 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 9 T + 128 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T + 142 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 14 T + 198 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.40509723935053206255780021760, −9.730747983450564972363666958319, −9.269461802051136599333512278135, −9.030349635085689639443277468277, −8.607425023989674458271902987602, −8.218481723170876935839658217475, −7.63377684795684572540054953690, −7.00515204683514271804791357069, −6.80217316125886040642651172131, −6.25170554024009670678048785667, −6.06325101073125175662248675889, −5.56507900355847824074066298697, −4.67093250850662199388310576228, −4.34525009015419484453738364839, −3.94670328982411401578264404479, −3.90135760216719732222356982045, −2.76636362403408315057351541959, −2.48187998105717011530260448552, −1.88554258137734670923623349210, −1.06121728191310280520996406203,
1.06121728191310280520996406203, 1.88554258137734670923623349210, 2.48187998105717011530260448552, 2.76636362403408315057351541959, 3.90135760216719732222356982045, 3.94670328982411401578264404479, 4.34525009015419484453738364839, 4.67093250850662199388310576228, 5.56507900355847824074066298697, 6.06325101073125175662248675889, 6.25170554024009670678048785667, 6.80217316125886040642651172131, 7.00515204683514271804791357069, 7.63377684795684572540054953690, 8.218481723170876935839658217475, 8.607425023989674458271902987602, 9.030349635085689639443277468277, 9.269461802051136599333512278135, 9.730747983450564972363666958319, 10.40509723935053206255780021760
