L(s) = 1 | − 7-s + 2·9-s − 8·23-s − 6·25-s − 4·29-s − 4·37-s + 49-s + 4·53-s − 2·63-s + 24·67-s + 16·71-s − 5·81-s + 8·107-s − 4·109-s − 12·113-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 8·161-s + 163-s + 167-s − 6·169-s + ⋯ |
L(s) = 1 | − 0.377·7-s + 2/3·9-s − 1.66·23-s − 6/5·25-s − 0.742·29-s − 0.657·37-s + 1/7·49-s + 0.549·53-s − 0.251·63-s + 2.93·67-s + 1.89·71-s − 5/9·81-s + 0.773·107-s − 0.383·109-s − 1.12·113-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.630·161-s + 0.0783·163-s + 0.0773·167-s − 0.461·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.479234028\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.479234028\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( 1 + T \) |
good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
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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
−7.891324420127686085550129195909, −7.59323168715670500590973969136, −6.95540896797002597284237909041, −6.78523961215961321298954807630, −6.14275733321769910969643004422, −5.83873273835049553454047245541, −5.30885500505738297133867999554, −4.89821029891924306193690564046, −4.15658020906930397374781788636, −3.77775849182394915622704046945, −3.61175651732633785166651233840, −2.63188894292257056466925258503, −2.10686711822222498297603419961, −1.60826206523482221383180587998, −0.51896389410269347045509565112,
0.51896389410269347045509565112, 1.60826206523482221383180587998, 2.10686711822222498297603419961, 2.63188894292257056466925258503, 3.61175651732633785166651233840, 3.77775849182394915622704046945, 4.15658020906930397374781788636, 4.89821029891924306193690564046, 5.30885500505738297133867999554, 5.83873273835049553454047245541, 6.14275733321769910969643004422, 6.78523961215961321298954807630, 6.95540896797002597284237909041, 7.59323168715670500590973969136, 7.891324420127686085550129195909