L(s) = 1 | − 4-s − 8·11-s + 16-s + 12·29-s + 16·31-s − 4·41-s + 8·44-s − 49-s − 16·59-s − 28·61-s − 64-s + 32·71-s + 16·79-s + 20·89-s + 12·101-s − 12·109-s − 12·116-s + 26·121-s − 16·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
L(s) = 1 | − 1/2·4-s − 2.41·11-s + 1/4·16-s + 2.22·29-s + 2.87·31-s − 0.624·41-s + 1.20·44-s − 1/7·49-s − 2.08·59-s − 3.58·61-s − 1/8·64-s + 3.79·71-s + 1.80·79-s + 2.11·89-s + 1.19·101-s − 1.14·109-s − 1.11·116-s + 2.36·121-s − 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9922500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9922500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.485992907\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.485992907\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
good | 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−8.620458847782727851776238916580, −8.536377015452642172596434025106, −7.982779834373795867493677771238, −7.78895582283566919281588038133, −7.76089125495258490407748005774, −6.97695876416584513340373372504, −6.49108766081539479049912208082, −6.20501446670060598078154713008, −6.00102859239398283377855456955, −5.10620748871220915446797362532, −5.06092795556312604874733531608, −4.63154069072538103516153316650, −4.55820511930585603030608886214, −3.63618017843396477109989720116, −3.22581543751014398934573656441, −2.65848868802849023646703715401, −2.63676871248544139266347366121, −1.84329559571027258917234335505, −0.993112740136791518600613631740, −0.45150450724187887527511920428,
0.45150450724187887527511920428, 0.993112740136791518600613631740, 1.84329559571027258917234335505, 2.63676871248544139266347366121, 2.65848868802849023646703715401, 3.22581543751014398934573656441, 3.63618017843396477109989720116, 4.55820511930585603030608886214, 4.63154069072538103516153316650, 5.06092795556312604874733531608, 5.10620748871220915446797362532, 6.00102859239398283377855456955, 6.20501446670060598078154713008, 6.49108766081539479049912208082, 6.97695876416584513340373372504, 7.76089125495258490407748005774, 7.78895582283566919281588038133, 7.982779834373795867493677771238, 8.536377015452642172596434025106, 8.620458847782727851776238916580