| L(s) = 1 | + 4.58e9·3-s + 1.09e12·4-s − 7.99e13·5-s + 8.89e18·9-s + 6.72e20·11-s + 5.04e21·12-s − 3.66e23·15-s + 1.20e24·16-s − 8.79e25·20-s + 4.90e25·23-s − 2.70e27·25-s − 1.49e28·27-s + 1.31e30·31-s + 3.08e30·33-s + 9.78e30·36-s + 3.79e31·37-s + 7.39e32·44-s − 7.11e32·45-s + 1.21e33·47-s + 5.54e33·48-s + 6.36e33·49-s − 1.74e34·53-s − 5.37e34·55-s − 3.92e35·59-s − 4.03e35·60-s + 1.32e36·64-s + 3.60e36·67-s + ⋯ |
| L(s) = 1 | + 1.31·3-s + 4-s − 0.838·5-s + 0.731·9-s + 11-s + 1.31·12-s − 1.10·15-s + 16-s − 0.838·20-s + 0.0285·23-s − 0.297·25-s − 0.352·27-s + 1.95·31-s + 1.31·33-s + 0.731·36-s + 1.64·37-s + 44-s − 0.613·45-s + 0.439·47-s + 1.31·48-s + 49-s − 0.570·53-s − 0.838·55-s − 1.50·59-s − 1.10·60-s + 64-s + 1.08·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(41-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+20) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{41}{2})\) |
\(\approx\) |
\(4.615934869\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.615934869\) |
| \(L(21)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - p^{20} T \) |
| good | 2 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 3 | \( 1 - 4588486927 T + p^{40} T^{2} \) |
| 5 | \( 1 + 79953996769249 T + p^{40} T^{2} \) |
| 7 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 13 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 17 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 19 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 23 | \( 1 - \)\(49\!\cdots\!27\)\( T + p^{40} T^{2} \) |
| 29 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 31 | \( 1 - \)\(13\!\cdots\!27\)\( T + p^{40} T^{2} \) |
| 37 | \( 1 - \)\(37\!\cdots\!27\)\( T + p^{40} T^{2} \) |
| 41 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 43 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 47 | \( 1 - \)\(12\!\cdots\!02\)\( T + p^{40} T^{2} \) |
| 53 | \( 1 + \)\(17\!\cdots\!98\)\( T + p^{40} T^{2} \) |
| 59 | \( 1 + \)\(39\!\cdots\!73\)\( T + p^{40} T^{2} \) |
| 61 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 67 | \( 1 - \)\(36\!\cdots\!27\)\( T + p^{40} T^{2} \) |
| 71 | \( 1 - \)\(13\!\cdots\!27\)\( T + p^{40} T^{2} \) |
| 73 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 79 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 83 | \( ( 1 - p^{20} T )( 1 + p^{20} T ) \) |
| 89 | \( 1 - \)\(98\!\cdots\!27\)\( T + p^{40} T^{2} \) |
| 97 | \( 1 + \)\(74\!\cdots\!73\)\( T + p^{40} T^{2} \) |
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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
−12.22942592997834753931100578369, −11.25368161287164774838689455571, −9.662004749316366103417688934477, −8.338536545838585110945256559545, −7.51676792574022980349601252172, −6.28389961972759009582312332430, −4.18374067369971676487017416877, −3.20302496066093463699303421819, −2.24711166295922783316038591410, −0.983739524020088034991977187664,
0.983739524020088034991977187664, 2.24711166295922783316038591410, 3.20302496066093463699303421819, 4.18374067369971676487017416877, 6.28389961972759009582312332430, 7.51676792574022980349601252172, 8.338536545838585110945256559545, 9.662004749316366103417688934477, 11.25368161287164774838689455571, 12.22942592997834753931100578369
