| L(s) = 1 | − 2·3-s + 4-s + 2·5-s + 3·9-s − 2·12-s − 4·15-s + 2·20-s + 2·23-s + 3·25-s − 4·27-s − 2·31-s + 3·36-s + 6·45-s − 2·47-s + 2·49-s − 2·53-s − 4·60-s − 64-s − 4·69-s − 6·75-s + 5·81-s + 2·92-s + 4·93-s + 3·100-s − 4·108-s + 2·113-s + 4·115-s + ⋯ |
| L(s) = 1 | − 2·3-s + 4-s + 2·5-s + 3·9-s − 2·12-s − 4·15-s + 2·20-s + 2·23-s + 3·25-s − 4·27-s − 2·31-s + 3·36-s + 6·45-s − 2·47-s + 2·49-s − 2·53-s − 4·60-s − 64-s − 4·69-s − 6·75-s + 5·81-s + 2·92-s + 4·93-s + 3·100-s − 4·108-s + 2·113-s + 4·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.253249032\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.253249032\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−9.656159348438347892927595392056, −9.537931299313523383273877423407, −8.950305856341564658739283209598, −8.778033178593796715735826706120, −7.84648844931249540853838992816, −7.23610872739760395777815749759, −7.20327146837067514980066063956, −6.69962481250731999581600669534, −6.42592339820266726956159402619, −6.06243175712680702213385968073, −5.72484151294612826459912536393, −5.30067903487659601179785626884, −4.89973313744619903837494091082, −4.76751469542819473074050634909, −3.90850182961347727927371126908, −3.20946499275907622107630563419, −2.66212566692108012495035884463, −1.86020521558123107755363071851, −1.70759946372457092048555556385, −1.02019953795764666445398560065,
1.02019953795764666445398560065, 1.70759946372457092048555556385, 1.86020521558123107755363071851, 2.66212566692108012495035884463, 3.20946499275907622107630563419, 3.90850182961347727927371126908, 4.76751469542819473074050634909, 4.89973313744619903837494091082, 5.30067903487659601179785626884, 5.72484151294612826459912536393, 6.06243175712680702213385968073, 6.42592339820266726956159402619, 6.69962481250731999581600669534, 7.20327146837067514980066063956, 7.23610872739760395777815749759, 7.84648844931249540853838992816, 8.778033178593796715735826706120, 8.950305856341564658739283209598, 9.537931299313523383273877423407, 9.656159348438347892927595392056
