| L(s) = 1 | + 1.04e6·2-s + 6.07e9·3-s + 1.09e12·4-s + 6.36e15·6-s + 1.15e18·8-s + 2.47e19·9-s + 4.50e20·11-s + 6.67e21·12-s + 1.20e24·16-s + 3.11e24·17-s + 2.59e25·18-s − 7.41e25·19-s + 4.72e26·22-s + 7.00e27·24-s + 9.09e27·25-s + 7.64e28·27-s + 1.26e30·32-s + 2.73e30·33-s + 3.26e30·34-s + 2.71e31·36-s − 7.77e31·38-s + 2.85e32·41-s − 9.26e32·43-s + 4.95e32·44-s + 7.34e33·48-s + 6.36e33·49-s + 9.53e33·50-s + ⋯ |
| L(s) = 1 | + 2-s + 1.74·3-s + 4-s + 1.74·6-s + 8-s + 2.03·9-s + 0.670·11-s + 1.74·12-s + 16-s + 0.767·17-s + 2.03·18-s − 1.97·19-s + 0.670·22-s + 1.74·24-s + 25-s + 1.80·27-s + 32-s + 1.16·33-s + 0.767·34-s + 2.03·36-s − 1.97·38-s + 1.58·41-s − 1.98·43-s + 0.670·44-s + 1.74·48-s + 49-s + 50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(41-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+20) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{41}{2})\) |
\(\approx\) |
\(9.163519468\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.163519468\) |
| \(L(21)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{20} T \) |
| good | 3 | \( 1 - 6074010274 T + p^{40} T^{2} \) |
| 5 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 7 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 11 | \( 1 - \)\(45\!\cdots\!74\)\( T + p^{40} T^{2} \) |
| 13 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 17 | \( 1 - \)\(31\!\cdots\!74\)\( T + p^{40} T^{2} \) |
| 19 | \( 1 + \)\(74\!\cdots\!26\)\( T + p^{40} T^{2} \) |
| 23 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 29 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 31 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 37 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 41 | \( 1 - \)\(28\!\cdots\!74\)\( T + p^{40} T^{2} \) |
| 43 | \( 1 + \)\(92\!\cdots\!98\)\( T + p^{40} T^{2} \) |
| 47 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 53 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 59 | \( 1 + \)\(49\!\cdots\!98\)\( T + p^{40} T^{2} \) |
| 61 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 67 | \( 1 + \)\(65\!\cdots\!26\)\( T + p^{40} T^{2} \) |
| 71 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 73 | \( 1 + \)\(73\!\cdots\!26\)\( T + p^{40} T^{2} \) |
| 79 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 83 | \( 1 - \)\(48\!\cdots\!74\)\( T + p^{40} T^{2} \) |
| 89 | \( 1 - \)\(18\!\cdots\!74\)\( T + p^{40} T^{2} \) |
| 97 | \( 1 + \)\(83\!\cdots\!98\)\( T + p^{40} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.43584524943433600033842998587, −12.38037830603487849341591736507, −10.49902902530454687279694806153, −8.951924686108265668227164934611, −7.74659680276416315582949594667, −6.45861680537687427087680969951, −4.50396560873164721928928611327, −3.52480903071920779321212172693, −2.49888384433323792077505047865, −1.45328371370504805446593303464,
1.45328371370504805446593303464, 2.49888384433323792077505047865, 3.52480903071920779321212172693, 4.50396560873164721928928611327, 6.45861680537687427087680969951, 7.74659680276416315582949594667, 8.951924686108265668227164934611, 10.49902902530454687279694806153, 12.38037830603487849341591736507, 13.43584524943433600033842998587
