| L(s) = 1 | − 2·3-s + 2·5-s + 7-s + 3·9-s + 11-s + 3·13-s − 4·15-s + 5·17-s − 7·19-s − 2·21-s − 3·23-s + 3·25-s − 4·27-s + 7·29-s − 31-s − 2·33-s + 2·35-s + 3·37-s − 6·39-s + 3·41-s + 8·43-s + 6·45-s + 3·47-s + 7·49-s − 10·51-s + 12·53-s + 2·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s + 0.377·7-s + 9-s + 0.301·11-s + 0.832·13-s − 1.03·15-s + 1.21·17-s − 1.60·19-s − 0.436·21-s − 0.625·23-s + 3/5·25-s − 0.769·27-s + 1.29·29-s − 0.179·31-s − 0.348·33-s + 0.338·35-s + 0.493·37-s − 0.960·39-s + 0.468·41-s + 1.21·43-s + 0.894·45-s + 0.437·47-s + 49-s − 1.40·51-s + 1.64·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16160400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16160400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.991287805\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.991287805\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 67 | $C_2$ | \( 1 - 16 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 3 T - 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 5 T + 8 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 7 T + 20 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 3 T - 28 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 3 T - 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 15 T + 154 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 3 T - 64 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 3 T - 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 11 T + 38 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 13 T + 72 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
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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
−8.497567595182442472127483938543, −8.494910501804390476884110801865, −7.74890340268782608617751992814, −7.56806846076379741500717265976, −7.04122317915188098145382334114, −6.70474618580469433972554903946, −6.21186672293638250179773881293, −6.08711984746274730597246522241, −5.65741416593099455815615141533, −5.52881154013073018092948689160, −4.93755883565379582843207128780, −4.45562296801687955525646812023, −4.06821239711296237091962497883, −3.98416875751325729197461905429, −3.03962634947120683570838337488, −2.75613455341640813542036081651, −1.95561095729193591857360997127, −1.76095255939046049382677383420, −0.955192796212620988039633999473, −0.67558602119566158094068470065,
0.67558602119566158094068470065, 0.955192796212620988039633999473, 1.76095255939046049382677383420, 1.95561095729193591857360997127, 2.75613455341640813542036081651, 3.03962634947120683570838337488, 3.98416875751325729197461905429, 4.06821239711296237091962497883, 4.45562296801687955525646812023, 4.93755883565379582843207128780, 5.52881154013073018092948689160, 5.65741416593099455815615141533, 6.08711984746274730597246522241, 6.21186672293638250179773881293, 6.70474618580469433972554903946, 7.04122317915188098145382334114, 7.56806846076379741500717265976, 7.74890340268782608617751992814, 8.494910501804390476884110801865, 8.497567595182442472127483938543
