| L(s) = 1 | + 2·3-s + 6·7-s + 4·19-s + 12·21-s + 6·23-s − 2·27-s − 12·31-s + 12·37-s − 12·41-s − 10·43-s + 6·47-s + 16·49-s + 8·57-s + 12·59-s + 12·61-s + 10·67-s + 12·69-s − 12·71-s − 8·73-s − 24·79-s − 81-s + 6·83-s + 12·89-s − 24·93-s − 8·97-s − 24·101-s + 6·103-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 2.26·7-s + 0.917·19-s + 2.61·21-s + 1.25·23-s − 0.384·27-s − 2.15·31-s + 1.97·37-s − 1.87·41-s − 1.52·43-s + 0.875·47-s + 16/7·49-s + 1.05·57-s + 1.56·59-s + 1.53·61-s + 1.22·67-s + 1.44·69-s − 1.42·71-s − 0.936·73-s − 2.70·79-s − 1/9·81-s + 0.658·83-s + 1.27·89-s − 2.48·93-s − 0.812·97-s − 2.38·101-s + 0.591·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.869063588\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.869063588\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 6 T + 20 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 12 T + 106 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 10 T + 84 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 76 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 10 T + 132 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 70 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 172 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−8.159184066526061706819791352320, −8.144658587975875353275521982021, −7.44970655023252587998989009464, −7.25931454523435601040537904145, −7.07439985985922664825036190158, −6.63488319512698082331812558322, −5.78377769160126327336632729521, −5.73144761451614474294172144249, −5.26172614377723480287251438802, −4.98639305406565939274551113236, −4.62323425191528013398175988362, −4.26737423715517994140563205991, −3.61765269783335511537919938283, −3.50961479417966787121207993830, −2.81559230919673045102680642592, −2.66923009663068985978238846299, −1.88880612314076948835207481263, −1.80367652396824163872344005306, −1.23497447079544740800187521831, −0.61137964985745817595391094137,
0.61137964985745817595391094137, 1.23497447079544740800187521831, 1.80367652396824163872344005306, 1.88880612314076948835207481263, 2.66923009663068985978238846299, 2.81559230919673045102680642592, 3.50961479417966787121207993830, 3.61765269783335511537919938283, 4.26737423715517994140563205991, 4.62323425191528013398175988362, 4.98639305406565939274551113236, 5.26172614377723480287251438802, 5.73144761451614474294172144249, 5.78377769160126327336632729521, 6.63488319512698082331812558322, 7.07439985985922664825036190158, 7.25931454523435601040537904145, 7.44970655023252587998989009464, 8.144658587975875353275521982021, 8.159184066526061706819791352320
