| L(s) = 1 | + 2·2-s − 3-s + 3·4-s − 2·6-s + 4·8-s + 11-s − 3·12-s + 5·16-s + 17-s + 19-s + 2·22-s − 4·24-s − 25-s + 27-s + 6·32-s − 33-s + 2·34-s + 2·38-s + 41-s + 43-s + 3·44-s − 5·48-s − 2·50-s − 51-s + 2·54-s − 57-s − 2·59-s + ⋯ |
| L(s) = 1 | + 2·2-s − 3-s + 3·4-s − 2·6-s + 4·8-s + 11-s − 3·12-s + 5·16-s + 17-s + 19-s + 2·22-s − 4·24-s − 25-s + 27-s + 6·32-s − 33-s + 2·34-s + 2·38-s + 41-s + 43-s + 3·44-s − 5·48-s − 2·50-s − 51-s + 2·54-s − 57-s − 2·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(4.926719019\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.926719019\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 19 | $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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 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
−8.942281177937295851963062239314, −8.363371304843866925656847384144, −7.984316189553014269252123803855, −7.53638095567513135107307524231, −7.19601705966550582409197381524, −7.11016547125511599469324015751, −6.29237138469354956020572672884, −6.21763174985182854420704634425, −5.76945437039590007902489668277, −5.75066527519856191970762705963, −5.04688727907165835542270659338, −4.93305104927964922664992850808, −4.21480995354495486681381330596, −4.15824983235975974127975237742, −3.55549073149014487289725356628, −3.11207990891171710293707960837, −2.78600325468325779502892754470, −2.18585957783322915029636424006, −1.28027676546130298915307731509, −1.27759764580490499074868799051,
1.27759764580490499074868799051, 1.28027676546130298915307731509, 2.18585957783322915029636424006, 2.78600325468325779502892754470, 3.11207990891171710293707960837, 3.55549073149014487289725356628, 4.15824983235975974127975237742, 4.21480995354495486681381330596, 4.93305104927964922664992850808, 5.04688727907165835542270659338, 5.75066527519856191970762705963, 5.76945437039590007902489668277, 6.21763174985182854420704634425, 6.29237138469354956020572672884, 7.11016547125511599469324015751, 7.19601705966550582409197381524, 7.53638095567513135107307524231, 7.984316189553014269252123803855, 8.363371304843866925656847384144, 8.942281177937295851963062239314
