| L(s) = 1 | − 4-s − 12·11-s + 16-s − 4·19-s + 16·31-s − 6·41-s + 12·44-s + 13·49-s + 12·59-s − 20·61-s − 64-s − 24·71-s + 4·76-s + 26·79-s + 18·89-s − 12·101-s + 32·109-s + 86·121-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 − 3.61·11-s + 1/4·16-s − 0.917·19-s + 2.87·31-s − 0.937·41-s + 1.80·44-s + 13/7·49-s + 1.56·59-s − 2.56·61-s − 1/8·64-s − 2.84·71-s + 0.458·76-s + 2.92·79-s + 1.90·89-s − 1.19·101-s + 3.06·109-s + 7.81·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16402500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16402500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8750363917\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8750363917\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 95 T^{2} + p^{2} T^{4} \) |
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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
−8.443543527109999486040773415037, −8.360513599788231649305551092323, −7.78861391901733425968005862456, −7.66664550777163495166842180272, −7.43454251005256896350504534642, −6.72223064171189385323875535870, −6.43968769814352673672335687940, −5.79981619360413301517388524391, −5.76711286000322917290163386923, −5.13452699826958485669962061430, −4.93109958198209828821033174466, −4.45990439335744863555543223553, −4.37840461121146647123840362926, −3.42651788310693317236485796022, −3.17253054456386988673960947603, −2.51259306415624349843873275129, −2.50947775301949639753157053890, −1.88648311828018344086146277706, −0.880704118397224503250104996619, −0.33230590500235985266084713897,
0.33230590500235985266084713897, 0.880704118397224503250104996619, 1.88648311828018344086146277706, 2.50947775301949639753157053890, 2.51259306415624349843873275129, 3.17253054456386988673960947603, 3.42651788310693317236485796022, 4.37840461121146647123840362926, 4.45990439335744863555543223553, 4.93109958198209828821033174466, 5.13452699826958485669962061430, 5.76711286000322917290163386923, 5.79981619360413301517388524391, 6.43968769814352673672335687940, 6.72223064171189385323875535870, 7.43454251005256896350504534642, 7.66664550777163495166842180272, 7.78861391901733425968005862456, 8.360513599788231649305551092323, 8.443543527109999486040773415037
