L(s) = 1 | − 4·2-s + 8·4-s − 5·7-s − 8·8-s + 9-s + 2·11-s + 20·14-s − 4·16-s − 4·18-s − 8·22-s − 8·23-s − 25-s − 40·28-s − 4·29-s + 32·32-s + 8·36-s − 2·43-s + 16·44-s + 32·46-s + 18·49-s + 4·50-s + 20·53-s + 40·56-s + 16·58-s − 5·63-s − 64·64-s + 16·67-s + ⋯ |
L(s) = 1 | − 2.82·2-s + 4·4-s − 1.88·7-s − 2.82·8-s + 1/3·9-s + 0.603·11-s + 5.34·14-s − 16-s − 0.942·18-s − 1.70·22-s − 1.66·23-s − 1/5·25-s − 7.55·28-s − 0.742·29-s + 5.65·32-s + 4/3·36-s − 0.304·43-s + 2.41·44-s + 4.71·46-s + 18/7·49-s + 0.565·50-s + 2.74·53-s + 5.34·56-s + 2.10·58-s − 0.629·63-s − 8·64-s + 1.95·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 159201 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159201 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 2 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.959163091125505196869204673189, −8.836427381447004052087018681826, −8.240904623337692395017705850814, −7.50551633645791376109359922005, −7.47848778675296589128891353641, −6.77741729329520342893941173390, −6.41173993136661837405396202601, −5.95880974340205553779958707744, −5.02464724211845145360498207900, −3.92167943470614491379352796776, −3.81591641511052084615077615666, −2.54192725156556063621144024653, −1.98299256211429317844363710869, −0.930874084897434565967858870729, 0,
0.930874084897434565967858870729, 1.98299256211429317844363710869, 2.54192725156556063621144024653, 3.81591641511052084615077615666, 3.92167943470614491379352796776, 5.02464724211845145360498207900, 5.95880974340205553779958707744, 6.41173993136661837405396202601, 6.77741729329520342893941173390, 7.47848778675296589128891353641, 7.50551633645791376109359922005, 8.240904623337692395017705850814, 8.836427381447004052087018681826, 8.959163091125505196869204673189