| L(s) = 1 | − 2·3-s − 4-s + 2·5-s + 9-s + 2·12-s − 13-s − 4·15-s + 17-s + 19-s − 2·20-s + 23-s + 3·25-s + 2·27-s + 2·31-s − 36-s + 37-s + 2·39-s + 41-s + 43-s + 2·45-s − 49-s − 2·51-s + 52-s − 2·53-s − 2·57-s + 59-s + 4·60-s + ⋯ |
| L(s) = 1 | − 2·3-s − 4-s + 2·5-s + 9-s + 2·12-s − 13-s − 4·15-s + 17-s + 19-s − 2·20-s + 23-s + 3·25-s + 2·27-s + 2·31-s − 36-s + 37-s + 2·39-s + 41-s + 43-s + 2·45-s − 49-s − 2·51-s + 52-s − 2·53-s − 2·57-s + 59-s + 4·60-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4060225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4060225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7158212527\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7158212527\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_2$ | \( 1 + T + T^{2} \) |
| 31 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 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_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 73 | $C_1$ | \( ( 1 - T )^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−9.676642283123131507476408982095, −9.201168240018315373142171739280, −9.129322024308572396869843617420, −8.224537453385269541028352786251, −8.205981176130642541653464924154, −7.51049586488008425549283797497, −6.87961465135138926115936289097, −6.54142234799390534556624658862, −6.32369784988611678914749843142, −5.86640189317603337710717035399, −5.48659295241841851870137711904, −5.16078195795805571652868798366, −4.88976290515679562469307580999, −4.75986483817534328164296509563, −3.98900634299216580074548649290, −2.98483254374812149878381685788, −2.82478551786765405845661030346, −2.21619469952025777608845809487, −0.990568801415586614304496020758, −0.986569720624933608347739662660,
0.986569720624933608347739662660, 0.990568801415586614304496020758, 2.21619469952025777608845809487, 2.82478551786765405845661030346, 2.98483254374812149878381685788, 3.98900634299216580074548649290, 4.75986483817534328164296509563, 4.88976290515679562469307580999, 5.16078195795805571652868798366, 5.48659295241841851870137711904, 5.86640189317603337710717035399, 6.32369784988611678914749843142, 6.54142234799390534556624658862, 6.87961465135138926115936289097, 7.51049586488008425549283797497, 8.205981176130642541653464924154, 8.224537453385269541028352786251, 9.129322024308572396869843617420, 9.201168240018315373142171739280, 9.676642283123131507476408982095
