L(s) = 1 | − 2·2-s + 3-s + 3·4-s − 5-s − 2·6-s − 2·7-s − 4·8-s + 2·10-s + 3·12-s + 4·14-s − 15-s + 5·16-s − 3·20-s − 2·21-s − 23-s − 4·24-s − 27-s − 6·28-s − 2·29-s + 2·30-s − 6·32-s + 2·35-s + 4·40-s − 2·41-s + 4·42-s − 43-s + 2·46-s + ⋯ |
L(s) = 1 | − 2·2-s + 3-s + 3·4-s − 5-s − 2·6-s − 2·7-s − 4·8-s + 2·10-s + 3·12-s + 4·14-s − 15-s + 5·16-s − 3·20-s − 2·21-s − 23-s − 4·24-s − 27-s − 6·28-s − 2·29-s + 2·30-s − 6·32-s + 2·35-s + 4·40-s − 2·41-s + 4·42-s − 43-s + 2·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1587600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1587600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1973987994\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1973987994\) |
\(L(1)\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−10.34274350950073446139120031952, −9.431738534360180948999203112528, −9.358703324789168012032938837631, −8.919677758309502858401307669867, −8.572349457134316848598707643106, −8.060199829049897515263824338643, −7.84487209970919216547975245298, −7.26131255256326681745662086456, −7.23543903994326109494923330283, −6.45370628656161309725169869624, −6.35867296920662831822305748199, −5.69619902306730047664280375369, −5.35243256168188764016349591502, −4.03952957339459151968367102394, −3.70100619853651069847666703394, −3.29145424545245558175570122383, −3.00505740290041440825946384635, −2.18057773115814201102304696471, −1.89000235761421345921146648137, −0.46453582372000240694536097617,
0.46453582372000240694536097617, 1.89000235761421345921146648137, 2.18057773115814201102304696471, 3.00505740290041440825946384635, 3.29145424545245558175570122383, 3.70100619853651069847666703394, 4.03952957339459151968367102394, 5.35243256168188764016349591502, 5.69619902306730047664280375369, 6.35867296920662831822305748199, 6.45370628656161309725169869624, 7.23543903994326109494923330283, 7.26131255256326681745662086456, 7.84487209970919216547975245298, 8.060199829049897515263824338643, 8.572349457134316848598707643106, 8.919677758309502858401307669867, 9.358703324789168012032938837631, 9.431738534360180948999203112528, 10.34274350950073446139120031952