| L(s) = 1 | + 27·3-s + 125·5-s + 729·9-s + 3.37e3·15-s + 9.39e3·17-s − 1.31e4·19-s + 1.46e4·23-s + 1.56e4·25-s + 1.96e4·27-s + 5.75e3·31-s + 9.11e4·45-s − 9.00e4·47-s + 1.17e5·49-s + 2.53e5·51-s + 8.86e4·53-s − 3.55e5·57-s − 3.25e5·61-s + 3.95e5·69-s + 4.21e5·75-s + 8.93e5·79-s + 5.31e5·81-s − 4.69e5·83-s + 1.17e6·85-s + 1.55e5·93-s − 1.64e6·95-s + 2.44e6·107-s + 7.73e5·109-s + ⋯ |
| L(s) = 1 | + 3-s + 5-s + 9-s + 15-s + 1.91·17-s − 1.92·19-s + 1.20·23-s + 25-s + 27-s + 0.193·31-s + 45-s − 0.867·47-s + 49-s + 1.91·51-s + 0.595·53-s − 1.92·57-s − 1.43·61-s + 1.20·69-s + 75-s + 1.81·79-s + 81-s − 0.821·83-s + 1.91·85-s + 0.193·93-s − 1.92·95-s + 1.99·107-s + 0.597·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(4.108661098\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.108661098\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{3} T \) |
| 5 | \( 1 - p^{3} T \) |
| good | 7 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 11 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 13 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 17 | \( 1 - 9394 T + p^{6} T^{2} \) |
| 19 | \( 1 + 13178 T + p^{6} T^{2} \) |
| 23 | \( 1 - 14654 T + p^{6} T^{2} \) |
| 29 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 31 | \( 1 - 5758 T + p^{6} T^{2} \) |
| 37 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 41 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 43 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 47 | \( 1 + 90034 T + p^{6} T^{2} \) |
| 53 | \( 1 - 88666 T + p^{6} T^{2} \) |
| 59 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 61 | \( 1 + 325798 T + p^{6} T^{2} \) |
| 67 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 71 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 73 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 79 | \( 1 - 893662 T + p^{6} T^{2} \) |
| 83 | \( 1 + 469546 T + p^{6} T^{2} \) |
| 89 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 97 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
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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
−10.72004016870564244948994258858, −10.01431017874961105960886652830, −9.118602561203071483327740871185, −8.280919450861712431426994694061, −7.14812206644735357353821717929, −6.04298349379793306255940655530, −4.77642480748247669492369478692, −3.38140166507491675674039460543, −2.29654160623007400600717103436, −1.16401360033574143875738755032,
1.16401360033574143875738755032, 2.29654160623007400600717103436, 3.38140166507491675674039460543, 4.77642480748247669492369478692, 6.04298349379793306255940655530, 7.14812206644735357353821717929, 8.280919450861712431426994694061, 9.118602561203071483327740871185, 10.01431017874961105960886652830, 10.72004016870564244948994258858
