| L(s) = 1 | − 3-s + 2·7-s − 13-s + 2·19-s − 2·21-s − 25-s + 27-s + 2·31-s + 2·37-s + 39-s + 43-s + 49-s − 2·57-s − 61-s + 67-s + 73-s + 75-s − 79-s − 81-s − 2·91-s − 2·93-s − 2·97-s + 2·103-s + 2·109-s − 2·111-s + 2·121-s + 127-s + ⋯ |
| L(s) = 1 | − 3-s + 2·7-s − 13-s + 2·19-s − 2·21-s − 25-s + 27-s + 2·31-s + 2·37-s + 39-s + 43-s + 49-s − 2·57-s − 61-s + 67-s + 73-s + 75-s − 79-s − 81-s − 2·91-s − 2·93-s − 2·97-s + 2·103-s + 2·109-s − 2·111-s + 2·121-s + 127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13307904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13307904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.399800868\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.399800868\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.702828164903063051553009003848, −8.512971740600568667733676054306, −8.063053297211937620720149226374, −7.73493964553281363988922690890, −7.39008687596838687981133557392, −7.37599989754003656175532149162, −6.49468763456651936031557517555, −6.32001415429653523283773479338, −5.73590366503616413405771091086, −5.58655315950096847431262977671, −4.96486442789649826618223603998, −4.94074912820256664861038156184, −4.33384677717227734615250663196, −4.33163571849155689124386288210, −3.35207373004196809703947895529, −2.96914197622087974226126921234, −2.39295452707180204993515215370, −1.98494456972485257460883014779, −1.15693359786272305363997907526, −0.893409089821771253732417731323,
0.893409089821771253732417731323, 1.15693359786272305363997907526, 1.98494456972485257460883014779, 2.39295452707180204993515215370, 2.96914197622087974226126921234, 3.35207373004196809703947895529, 4.33163571849155689124386288210, 4.33384677717227734615250663196, 4.94074912820256664861038156184, 4.96486442789649826618223603998, 5.58655315950096847431262977671, 5.73590366503616413405771091086, 6.32001415429653523283773479338, 6.49468763456651936031557517555, 7.37599989754003656175532149162, 7.39008687596838687981133557392, 7.73493964553281363988922690890, 8.063053297211937620720149226374, 8.512971740600568667733676054306, 8.702828164903063051553009003848
