| L(s) = 1 | + 2-s + 2·3-s + 2·6-s − 8-s + 9-s − 2·11-s − 16-s − 17-s + 18-s + 2·19-s − 2·22-s − 2·24-s − 2·27-s − 4·33-s − 34-s + 2·38-s + 41-s + 2·43-s − 2·48-s + 2·49-s − 2·51-s − 2·54-s + 4·57-s + 59-s + 64-s − 4·66-s − 67-s + ⋯ |
| L(s) = 1 | + 2-s + 2·3-s + 2·6-s − 8-s + 9-s − 2·11-s − 16-s − 17-s + 18-s + 2·19-s − 2·22-s − 2·24-s − 2·27-s − 4·33-s − 34-s + 2·38-s + 41-s + 2·43-s − 2·48-s + 2·49-s − 2·51-s − 2·54-s + 4·57-s + 59-s + 64-s − 4·66-s − 67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.074903943\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.074903943\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_1$ | \( ( 1 + T )^{4} \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 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
−9.114906605203099348350691795048, −8.382015224424303578515150486612, −8.214388630660322052894054126541, −7.79762057789068554485671239146, −7.52537154744642913452819390419, −7.16848059185293088992999408260, −6.81476535246409485699317389686, −5.95113986075958895750743345774, −5.78220173039908980144772048898, −5.35735827146484612942498489823, −5.28638264262488928653287288893, −4.54594366836949936912247901336, −4.03281896099082884616828639694, −3.98462332578615772159293043261, −3.24833178938098261104631913211, −2.87706574796491818910088509642, −2.72319435301908874115882307727, −2.42421003802135871988035216899, −1.86938424870237193609413054733, −0.71116573236821241728007108736,
0.71116573236821241728007108736, 1.86938424870237193609413054733, 2.42421003802135871988035216899, 2.72319435301908874115882307727, 2.87706574796491818910088509642, 3.24833178938098261104631913211, 3.98462332578615772159293043261, 4.03281896099082884616828639694, 4.54594366836949936912247901336, 5.28638264262488928653287288893, 5.35735827146484612942498489823, 5.78220173039908980144772048898, 5.95113986075958895750743345774, 6.81476535246409485699317389686, 7.16848059185293088992999408260, 7.52537154744642913452819390419, 7.79762057789068554485671239146, 8.214388630660322052894054126541, 8.382015224424303578515150486612, 9.114906605203099348350691795048
