| L(s) = 1 | − 5-s − 2·7-s + 9-s + 2·11-s + 19-s + 25-s + 2·35-s − 2·43-s − 45-s + 49-s − 3·53-s − 2·55-s − 2·63-s − 4·77-s + 3·79-s − 2·83-s − 95-s + 3·97-s + 2·99-s + 3·121-s − 2·125-s + 127-s + 131-s − 2·133-s + 137-s + 139-s + 149-s + ⋯ |
| L(s) = 1 | − 5-s − 2·7-s + 9-s + 2·11-s + 19-s + 25-s + 2·35-s − 2·43-s − 45-s + 49-s − 3·53-s − 2·55-s − 2·63-s − 4·77-s + 3·79-s − 2·83-s − 95-s + 3·97-s + 2·99-s + 3·121-s − 2·125-s + 127-s + 131-s − 2·133-s + 137-s + 139-s + 149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11182336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11182336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9652838598\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9652838598\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 43 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 71 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
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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
−9.131009534053220939561174464906, −8.781709953879737035262689963189, −8.106058885971312147726734417527, −7.893658345588534422232579627636, −7.45191202393599930272302479161, −7.02186097365728615672213171817, −6.63350743497419124202106247715, −6.43345799203336557288264208870, −6.40826105172078709610623400627, −5.66319321749649679007861006794, −5.09324899292741106140592203599, −4.63596351113743426226949989137, −4.33183717411393938849532959941, −3.76412889198418440304219515925, −3.40762661930304385902763075025, −3.32729503557027322658926649766, −2.82994591263405118069470157563, −1.79545191185697801519104198143, −1.44126639225351194546092073727, −0.62839135175571514933065219040,
0.62839135175571514933065219040, 1.44126639225351194546092073727, 1.79545191185697801519104198143, 2.82994591263405118069470157563, 3.32729503557027322658926649766, 3.40762661930304385902763075025, 3.76412889198418440304219515925, 4.33183717411393938849532959941, 4.63596351113743426226949989137, 5.09324899292741106140592203599, 5.66319321749649679007861006794, 6.40826105172078709610623400627, 6.43345799203336557288264208870, 6.63350743497419124202106247715, 7.02186097365728615672213171817, 7.45191202393599930272302479161, 7.893658345588534422232579627636, 8.106058885971312147726734417527, 8.781709953879737035262689963189, 9.131009534053220939561174464906
