| L(s) = 1 | + 2-s + 3-s + 2·5-s + 6-s − 8-s + 2·10-s + 11-s − 2·13-s + 2·15-s − 16-s + 2·17-s + 22-s − 23-s − 24-s + 3·25-s − 2·26-s − 27-s + 29-s + 2·30-s − 2·31-s + 33-s + 2·34-s − 37-s − 2·39-s − 2·40-s − 43-s − 46-s + ⋯ |
| L(s) = 1 | + 2-s + 3-s + 2·5-s + 6-s − 8-s + 2·10-s + 11-s − 2·13-s + 2·15-s − 16-s + 2·17-s + 22-s − 23-s − 24-s + 3·25-s − 2·26-s − 27-s + 29-s + 2·30-s − 2·31-s + 33-s + 2·34-s − 37-s − 2·39-s − 2·40-s − 43-s − 46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2433600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2433600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.158090140\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.158090140\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 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_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−9.674516759347995796375711158915, −9.451421639569907943777122926768, −9.249031244255154979388248272133, −8.687983489854475601254655562890, −8.389660416847555959583640864131, −7.87476412151513173883305156753, −7.32942841300325604525714632397, −6.84124776365321334377295082482, −6.66224575182956559068573416486, −5.94181349516306851231751597713, −5.52729957483340176381691928295, −5.39258316969940151966382770308, −5.09558883499970559186821985388, −4.30435386369030673407170817951, −3.93182701437690341874619595012, −3.13643494606143621371168042436, −3.12363436042882851747744328296, −2.28910229991004554029151424999, −2.08845433778762338911688563959, −1.30346326580925621012273945137,
1.30346326580925621012273945137, 2.08845433778762338911688563959, 2.28910229991004554029151424999, 3.12363436042882851747744328296, 3.13643494606143621371168042436, 3.93182701437690341874619595012, 4.30435386369030673407170817951, 5.09558883499970559186821985388, 5.39258316969940151966382770308, 5.52729957483340176381691928295, 5.94181349516306851231751597713, 6.66224575182956559068573416486, 6.84124776365321334377295082482, 7.32942841300325604525714632397, 7.87476412151513173883305156753, 8.389660416847555959583640864131, 8.687983489854475601254655562890, 9.249031244255154979388248272133, 9.451421639569907943777122926768, 9.674516759347995796375711158915
