| L(s) = 1 | + 3-s + 7-s + 3·9-s − 11-s − 2·13-s + 6·17-s + 2·19-s + 21-s − 6·23-s + 5·25-s + 8·27-s + 18·29-s − 4·31-s − 33-s − 2·37-s − 2·39-s − 12·41-s + 8·43-s − 6·47-s − 6·49-s + 6·51-s + 2·57-s − 3·59-s − 11·61-s + 3·63-s + 11·67-s − 6·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 9-s − 0.301·11-s − 0.554·13-s + 1.45·17-s + 0.458·19-s + 0.218·21-s − 1.25·23-s + 25-s + 1.53·27-s + 3.34·29-s − 0.718·31-s − 0.174·33-s − 0.328·37-s − 0.320·39-s − 1.87·41-s + 1.21·43-s − 0.875·47-s − 6/7·49-s + 0.840·51-s + 0.264·57-s − 0.390·59-s − 1.40·61-s + 0.377·63-s + 1.34·67-s − 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1517824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1517824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.395771613\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.395771613\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| 11 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 5 T - 54 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 18 T + 235 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
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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
−9.815991239653955782599352048980, −9.746976082664709018941615809743, −9.090267695284946990880242048174, −8.460082282030837681224886975991, −8.412662959080484984361178184546, −7.897492564547425010964100935566, −7.57374038880333160460192713466, −7.12505369747214846911742528632, −6.53147148132224377359485810026, −6.43598085356043037046923057859, −5.66571763480823976660190355906, −5.06407446064972615015910222858, −4.76231918368796292041647115324, −4.52381749628072942167802289990, −3.67218872598461066213710518050, −3.21685769585488133189518104713, −2.85905336288879770823024735656, −2.13929464396301414031242018120, −1.45226463344471417784925838315, −0.828614476607184150786913457889,
0.828614476607184150786913457889, 1.45226463344471417784925838315, 2.13929464396301414031242018120, 2.85905336288879770823024735656, 3.21685769585488133189518104713, 3.67218872598461066213710518050, 4.52381749628072942167802289990, 4.76231918368796292041647115324, 5.06407446064972615015910222858, 5.66571763480823976660190355906, 6.43598085356043037046923057859, 6.53147148132224377359485810026, 7.12505369747214846911742528632, 7.57374038880333160460192713466, 7.897492564547425010964100935566, 8.412662959080484984361178184546, 8.460082282030837681224886975991, 9.090267695284946990880242048174, 9.746976082664709018941615809743, 9.815991239653955782599352048980
