| L(s) = 1 | − 2·2-s + 3·4-s + 2·5-s − 2·7-s − 4·8-s − 4·10-s + 6·11-s − 2·13-s + 4·14-s + 5·16-s − 2·17-s + 2·19-s + 6·20-s − 12·22-s + 6·23-s − 2·25-s + 4·26-s − 6·28-s + 8·29-s − 4·31-s − 6·32-s + 4·34-s − 4·35-s − 12·37-s − 4·38-s − 8·40-s + 12·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 0.894·5-s − 0.755·7-s − 1.41·8-s − 1.26·10-s + 1.80·11-s − 0.554·13-s + 1.06·14-s + 5/4·16-s − 0.485·17-s + 0.458·19-s + 1.34·20-s − 2.55·22-s + 1.25·23-s − 2/5·25-s + 0.784·26-s − 1.13·28-s + 1.48·29-s − 0.718·31-s − 1.06·32-s + 0.685·34-s − 0.676·35-s − 1.97·37-s − 0.648·38-s − 1.26·40-s + 1.87·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5731236 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5731236 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.837406505\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.837406505\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T - 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_4$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 12 T + 90 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 8 T + 118 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 14 T + 138 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 158 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 16 T + 190 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 6 T + 162 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 14 T + 238 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
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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
−8.991295892472330142328999481590, −8.858885999448037013391045226848, −8.715010433521129373296102434571, −8.225012384596411764012013771353, −7.44464356226274058360973970324, −7.22325951581538230390779888314, −6.87484897037954637882134003639, −6.72237546101772004308449263692, −6.03441320911308192521077366486, −6.01899377163174968850444715338, −5.24405013490874731874129327005, −5.07228105561494212680214631170, −4.12144236108976087326420574675, −3.87763298788434150674352282358, −3.27963107636739416584393614623, −2.77473240337934365991383204539, −2.11753775817860732627324061317, −1.94437144593608726573366630229, −0.941750525556653715752685405721, −0.75755249820635417138551374004,
0.75755249820635417138551374004, 0.941750525556653715752685405721, 1.94437144593608726573366630229, 2.11753775817860732627324061317, 2.77473240337934365991383204539, 3.27963107636739416584393614623, 3.87763298788434150674352282358, 4.12144236108976087326420574675, 5.07228105561494212680214631170, 5.24405013490874731874129327005, 6.01899377163174968850444715338, 6.03441320911308192521077366486, 6.72237546101772004308449263692, 6.87484897037954637882134003639, 7.22325951581538230390779888314, 7.44464356226274058360973970324, 8.225012384596411764012013771353, 8.715010433521129373296102434571, 8.858885999448037013391045226848, 8.991295892472330142328999481590
