| L(s) = 1 | + 4.30e7·3-s − 4.39e13·7-s + 1.85e15·9-s − 1.20e18·13-s + 5.03e20·19-s − 1.89e21·21-s + 2.32e22·25-s + 7.97e22·27-s + 1.00e24·31-s + 2.11e25·37-s − 5.20e25·39-s − 1.48e26·43-s + 8.28e26·49-s + 2.16e28·57-s − 3.95e28·61-s − 8.14e28·63-s + 2.82e29·67-s + 5.25e29·73-s + 1.00e30·75-s + 2.56e30·79-s + 3.43e30·81-s + 5.31e31·91-s + 4.32e31·93-s + 1.21e32·97-s + 2.24e32·103-s − 7.92e32·109-s + 9.11e32·111-s + ⋯ |
| L(s) = 1 | + 3-s − 1.32·7-s + 9-s − 1.81·13-s + 1.74·19-s − 1.32·21-s + 25-s + 27-s + 1.38·31-s + 1.71·37-s − 1.81·39-s − 1.08·43-s + 0.750·49-s + 1.74·57-s − 1.07·61-s − 1.32·63-s + 1.71·67-s + 0.808·73-s + 75-s + 1.11·79-s + 81-s + 2.40·91-s + 1.38·93-s + 1.97·97-s + 1.40·103-s − 1.99·109-s + 1.71·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{33}{2})\) |
\(\approx\) |
\(2.599533439\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.599533439\) |
| \(L(17)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{16} T \) |
| good | 5 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 7 | \( 1 + 43964556588286 T + p^{32} T^{2} \) |
| 11 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 13 | \( 1 + 1208758584266418046 T + p^{32} T^{2} \) |
| 17 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 19 | \( 1 - \)\(50\!\cdots\!74\)\( T + p^{32} T^{2} \) |
| 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 | \( 1 - \)\(21\!\cdots\!82\)\( T + p^{32} T^{2} \) |
| 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 | \( 1 + \)\(39\!\cdots\!26\)\( T + p^{32} T^{2} \) |
| 67 | \( 1 - \)\(28\!\cdots\!54\)\( T + p^{32} T^{2} \) |
| 71 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 73 | \( 1 - \)\(52\!\cdots\!22\)\( T + p^{32} T^{2} \) |
| 79 | \( 1 - \)\(25\!\cdots\!42\)\( T + p^{32} T^{2} \) |
| 83 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 89 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 97 | \( 1 - \)\(12\!\cdots\!14\)\( T + p^{32} T^{2} \) |
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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
−13.20384557762442466319624269558, −12.13681800047070812321352559064, −9.967047740574268657199524410569, −9.408973372920976155663240219625, −7.75599911213590882886368916999, −6.70023434420368690185780181201, −4.83876677316322975726048884476, −3.27144034030812236780421338416, −2.52513248892095006353428200318, −0.77554958530612896910283421344,
0.77554958530612896910283421344, 2.52513248892095006353428200318, 3.27144034030812236780421338416, 4.83876677316322975726048884476, 6.70023434420368690185780181201, 7.75599911213590882886368916999, 9.408973372920976155663240219625, 9.967047740574268657199524410569, 12.13681800047070812321352559064, 13.20384557762442466319624269558
