| L(s) = 1 | − 5-s − 7-s + 11-s + 25-s + 2·29-s − 31-s + 35-s + 49-s + 2·53-s − 55-s − 2·59-s − 2·73-s − 77-s + 2·79-s + 83-s + 97-s − 101-s + 2·103-s − 2·107-s + 121-s − 2·125-s + 127-s + 131-s + 137-s + 139-s − 2·145-s + 149-s + ⋯ |
| L(s) = 1 | − 5-s − 7-s + 11-s + 25-s + 2·29-s − 31-s + 35-s + 49-s + 2·53-s − 55-s − 2·59-s − 2·73-s − 77-s + 2·79-s + 83-s + 97-s − 101-s + 2·103-s − 2·107-s + 121-s − 2·125-s + 127-s + 131-s + 137-s + 139-s − 2·145-s + 149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9343203259\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9343203259\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 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_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 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
−9.236351314620202215975435214645, −8.905895298087675900353140398441, −8.462242980186701268989682456803, −8.275498004458781677615796878207, −7.60017406881604383759233798517, −7.30115882129022975819495134651, −7.10421947644500331215620635119, −6.44932738062071186965230047666, −6.34161385326774547526252487351, −5.97292052010385556696729598962, −5.25541716917249270647105251496, −4.95945884258262340209743730382, −4.29706643833271862033738187812, −4.15329954918298847120546764565, −3.56702534412855749435095641453, −3.24853461402140327716126459240, −2.79012740984902619266998920516, −2.19776527116221247972419327822, −1.36597221637941752799478849235, −0.68953420565427865974920611317,
0.68953420565427865974920611317, 1.36597221637941752799478849235, 2.19776527116221247972419327822, 2.79012740984902619266998920516, 3.24853461402140327716126459240, 3.56702534412855749435095641453, 4.15329954918298847120546764565, 4.29706643833271862033738187812, 4.95945884258262340209743730382, 5.25541716917249270647105251496, 5.97292052010385556696729598962, 6.34161385326774547526252487351, 6.44932738062071186965230047666, 7.10421947644500331215620635119, 7.30115882129022975819495134651, 7.60017406881604383759233798517, 8.275498004458781677615796878207, 8.462242980186701268989682456803, 8.905895298087675900353140398441, 9.236351314620202215975435214645
