| L(s) = 1 | + 4.29e9·4-s + 6.51e13·7-s + 1.08e18·13-s + 1.84e19·16-s − 7.82e18·19-s + 2.32e22·25-s + 2.79e23·28-s + 1.00e24·31-s + 3.79e23·37-s − 1.48e26·43-s + 3.14e27·49-s + 4.66e27·52-s − 3.38e28·61-s + 7.92e28·64-s − 2.88e29·67-s + 7.67e29·73-s − 3.35e28·76-s − 4.59e30·79-s + 7.07e31·91-s − 7.80e31·97-s + 9.99e31·100-s − 3.10e32·103-s − 7.92e32·109-s + 1.20e33·112-s + ⋯ |
| L(s) = 1 | + 4-s + 1.96·7-s + 1.63·13-s + 16-s − 0.0271·19-s + 25-s + 1.96·28-s + 1.38·31-s + 0.0307·37-s − 1.08·43-s + 2.84·49-s + 1.63·52-s − 0.922·61-s + 64-s − 1.74·67-s + 1.18·73-s − 0.0271·76-s − 1.99·79-s + 3.20·91-s − 1.27·97-s + 100-s − 1.93·103-s − 1.99·109-s + 1.96·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{33}{2})\) |
\(\approx\) |
\(5.076197926\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.076197926\) |
| \(L(17)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 5 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 7 | \( 1 - 65151959156159 T + p^{32} T^{2} \) |
| 11 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 13 | \( 1 - 1086577182062443007 T + p^{32} T^{2} \) |
| 17 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 19 | \( 1 + 7821153231347348161 T + p^{32} T^{2} \) |
| 23 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 29 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 31 | \( 1 - \)\(10\!\cdots\!54\)\( T + p^{32} T^{2} \) |
| 37 | \( 1 - \)\(37\!\cdots\!79\)\( T + p^{32} T^{2} \) |
| 41 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 43 | \( 1 + \)\(14\!\cdots\!46\)\( T + p^{32} T^{2} \) |
| 47 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 53 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 59 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 61 | \( 1 + \)\(33\!\cdots\!21\)\( T + p^{32} T^{2} \) |
| 67 | \( 1 + \)\(28\!\cdots\!13\)\( T + p^{32} T^{2} \) |
| 71 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 73 | \( 1 - \)\(76\!\cdots\!79\)\( T + p^{32} T^{2} \) |
| 79 | \( 1 + \)\(45\!\cdots\!81\)\( T + p^{32} T^{2} \) |
| 83 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 89 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 97 | \( 1 + \)\(78\!\cdots\!81\)\( T + p^{32} T^{2} \) |
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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
−11.20085164346306139421572747436, −10.54867664065471059972584706415, −8.600455030227982051731426777158, −7.896149673731215458379368956222, −6.63231333155764189319004480969, −5.47941546004681315824342443304, −4.29628935136360093449188767137, −2.88619232268360444504620883374, −1.62316227353450859307629578012, −1.12187651927639695018012584417,
1.12187651927639695018012584417, 1.62316227353450859307629578012, 2.88619232268360444504620883374, 4.29628935136360093449188767137, 5.47941546004681315824342443304, 6.63231333155764189319004480969, 7.896149673731215458379368956222, 8.600455030227982051731426777158, 10.54867664065471059972584706415, 11.20085164346306139421572747436
