L(s) = 1 | + 2·2-s + 2·4-s − 2·11-s − 4·16-s + 2·17-s − 2·19-s − 4·22-s − 25-s − 8·32-s + 4·34-s − 4·38-s − 2·43-s − 4·44-s + 11·49-s − 2·50-s + 16·59-s − 8·64-s + 16·67-s + 4·68-s − 22·73-s − 4·76-s − 24·83-s − 4·86-s + 12·89-s − 20·97-s + 22·98-s − 2·100-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 4-s − 0.603·11-s − 16-s + 0.485·17-s − 0.458·19-s − 0.852·22-s − 1/5·25-s − 1.41·32-s + 0.685·34-s − 0.648·38-s − 0.304·43-s − 0.603·44-s + 11/7·49-s − 0.282·50-s + 2.08·59-s − 64-s + 1.95·67-s + 0.485·68-s − 2.57·73-s − 0.458·76-s − 2.63·83-s − 0.431·86-s + 1.27·89-s − 2.03·97-s + 2.22·98-s − 1/5·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−7.46607717556725394550853600860, −6.85073886350288365363080648917, −6.81976651111915002122674082793, −6.12241751609850681792173625996, −5.68404754505961415392399642199, −5.34668146252658900228619106104, −5.12252201851867645807416891194, −4.32311343904703138277137352813, −4.10417783076595364297890095109, −3.67840796781373175574076394185, −2.97029581237660750527806680564, −2.61687998381550726785939862479, −2.09376317548126210224444667076, −1.18431344333758512049731147264, 0,
1.18431344333758512049731147264, 2.09376317548126210224444667076, 2.61687998381550726785939862479, 2.97029581237660750527806680564, 3.67840796781373175574076394185, 4.10417783076595364297890095109, 4.32311343904703138277137352813, 5.12252201851867645807416891194, 5.34668146252658900228619106104, 5.68404754505961415392399642199, 6.12241751609850681792173625996, 6.81976651111915002122674082793, 6.85073886350288365363080648917, 7.46607717556725394550853600860