| L(s) = 1 | − 8.59e7·3-s + 4.29e9·4-s − 9.72e9·5-s + 5.53e15·9-s + 4.59e16·11-s − 3.69e17·12-s + 8.36e17·15-s + 1.84e19·16-s − 4.17e19·20-s + 3.65e21·23-s − 2.31e22·25-s − 3.16e23·27-s − 1.00e24·31-s − 3.95e24·33-s + 2.37e25·36-s + 1.77e25·37-s + 1.97e26·44-s − 5.38e25·45-s − 1.02e27·47-s − 1.58e27·48-s + 1.10e27·49-s + 4.02e27·53-s − 4.47e26·55-s + 3.35e28·59-s + 3.59e27·60-s + 7.92e28·64-s − 1.53e29·67-s + ⋯ |
| L(s) = 1 | − 1.99·3-s + 4-s − 0.0637·5-s + 2.98·9-s + 11-s − 1.99·12-s + 0.127·15-s + 16-s − 0.0637·20-s + 0.596·23-s − 0.995·25-s − 3.97·27-s − 1.38·31-s − 1.99·33-s + 2.98·36-s + 1.43·37-s + 44-s − 0.190·45-s − 1.80·47-s − 1.99·48-s + 49-s + 1.03·53-s − 0.0637·55-s + 1.55·59-s + 0.127·60-s + 64-s − 0.931·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{33}{2})\) |
\(\approx\) |
\(1.464993945\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.464993945\) |
| \(L(17)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - p^{16} T \) |
| good | 2 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 3 | \( 1 + 85968833 T + p^{32} T^{2} \) |
| 5 | \( 1 + 9728091649 T + p^{32} T^{2} \) |
| 7 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 13 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 17 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 19 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 23 | \( 1 - \)\(36\!\cdots\!47\)\( T + p^{32} T^{2} \) |
| 29 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 31 | \( 1 + \)\(10\!\cdots\!13\)\( T + p^{32} T^{2} \) |
| 37 | \( 1 - \)\(17\!\cdots\!07\)\( T + p^{32} T^{2} \) |
| 41 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 43 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 47 | \( 1 + \)\(10\!\cdots\!58\)\( T + p^{32} T^{2} \) |
| 53 | \( 1 - \)\(40\!\cdots\!42\)\( T + p^{32} T^{2} \) |
| 59 | \( 1 - \)\(33\!\cdots\!07\)\( T + p^{32} T^{2} \) |
| 61 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 67 | \( 1 + \)\(15\!\cdots\!13\)\( T + p^{32} T^{2} \) |
| 71 | \( 1 - \)\(70\!\cdots\!67\)\( T + p^{32} T^{2} \) |
| 73 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 79 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 83 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 89 | \( 1 + \)\(30\!\cdots\!53\)\( T + p^{32} T^{2} \) |
| 97 | \( 1 + \)\(40\!\cdots\!33\)\( T + p^{32} T^{2} \) |
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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
−12.79681991640641744096439676747, −11.69815852947917030429194207633, −11.12793170999355942816611734265, −9.862825479993612420324049217911, −7.31160544412265240312694550235, −6.40906376990926192762832537484, −5.46735563983969898430666425266, −3.96812195673020274630076984686, −1.75624228896501795476104167445, −0.72381146104502178281250642626,
0.72381146104502178281250642626, 1.75624228896501795476104167445, 3.96812195673020274630076984686, 5.46735563983969898430666425266, 6.40906376990926192762832537484, 7.31160544412265240312694550235, 9.862825479993612420324049217911, 11.12793170999355942816611734265, 11.69815852947917030429194207633, 12.79681991640641744096439676747
