| L(s) = 1 | − 8.38e6·2-s + 1.88e11·3-s + 7.03e13·4-s − 1.57e18·6-s − 5.90e20·8-s + 2.65e22·9-s + 2.79e23·11-s + 1.32e25·12-s + 4.95e27·16-s − 3.89e28·17-s − 2.22e29·18-s + 1.75e29·19-s − 2.34e30·22-s − 1.11e32·24-s + 1.42e32·25-s + 3.32e33·27-s − 4.15e34·32-s + 5.26e34·33-s + 3.27e35·34-s + 1.86e36·36-s − 1.47e36·38-s + 2.24e37·41-s + 4.31e37·43-s + 1.96e37·44-s + 9.31e38·48-s + 7.49e38·49-s − 1.19e39·50-s + ⋯ |
| L(s) = 1 | − 2-s + 1.99·3-s + 4-s − 1.99·6-s − 8-s + 2.99·9-s + 0.312·11-s + 1.99·12-s + 16-s − 1.95·17-s − 2.99·18-s + 0.681·19-s − 0.312·22-s − 1.99·24-s + 25-s + 3.98·27-s − 32-s + 0.624·33-s + 1.95·34-s + 2.99·36-s − 0.681·38-s + 1.81·41-s + 1.16·43-s + 0.312·44-s + 1.99·48-s + 49-s − 50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(47-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+23) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{47}{2})\) |
\(\approx\) |
\(3.509725617\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.509725617\) |
| \(L(24)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{23} T \) |
| good | 3 | \( 1 - 188152336462 T + p^{46} T^{2} \) |
| 5 | \( ( 1 - p^{23} T )( 1 + p^{23} T ) \) |
| 7 | \( ( 1 - p^{23} T )( 1 + p^{23} T ) \) |
| 11 | \( 1 - \)\(27\!\cdots\!94\)\( T + p^{46} T^{2} \) |
| 13 | \( ( 1 - p^{23} T )( 1 + p^{23} T ) \) |
| 17 | \( 1 + \)\(38\!\cdots\!82\)\( T + p^{46} T^{2} \) |
| 19 | \( 1 - \)\(17\!\cdots\!86\)\( T + p^{46} T^{2} \) |
| 23 | \( ( 1 - p^{23} T )( 1 + p^{23} T ) \) |
| 29 | \( ( 1 - p^{23} T )( 1 + p^{23} T ) \) |
| 31 | \( ( 1 - p^{23} T )( 1 + p^{23} T ) \) |
| 37 | \( ( 1 - p^{23} T )( 1 + p^{23} T ) \) |
| 41 | \( 1 - \)\(22\!\cdots\!94\)\( T + p^{46} T^{2} \) |
| 43 | \( 1 - \)\(43\!\cdots\!86\)\( T + p^{46} T^{2} \) |
| 47 | \( ( 1 - p^{23} T )( 1 + p^{23} T ) \) |
| 53 | \( ( 1 - p^{23} T )( 1 + p^{23} T ) \) |
| 59 | \( 1 + \)\(99\!\cdots\!42\)\( T + p^{46} T^{2} \) |
| 61 | \( ( 1 - p^{23} T )( 1 + p^{23} T ) \) |
| 67 | \( 1 - \)\(19\!\cdots\!98\)\( T + p^{46} T^{2} \) |
| 71 | \( ( 1 - p^{23} T )( 1 + p^{23} T ) \) |
| 73 | \( 1 + \)\(90\!\cdots\!78\)\( T + p^{46} T^{2} \) |
| 79 | \( ( 1 - p^{23} T )( 1 + p^{23} T ) \) |
| 83 | \( 1 - \)\(17\!\cdots\!62\)\( T + p^{46} T^{2} \) |
| 89 | \( 1 - \)\(17\!\cdots\!26\)\( T + p^{46} T^{2} \) |
| 97 | \( 1 + \)\(79\!\cdots\!46\)\( T + p^{46} T^{2} \) |
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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
−12.69549999626950664910485252938, −10.77745306640211080768639104426, −9.371007932926603702345524282267, −8.847504543034012959748680300118, −7.70120378742242110037791567344, −6.74093461764243896536719894486, −4.20772431751425981373654616501, −2.89220035690391433096944765543, −2.12458922328572519259860851804, −0.977449276036223243555157263877,
0.977449276036223243555157263877, 2.12458922328572519259860851804, 2.89220035690391433096944765543, 4.20772431751425981373654616501, 6.74093461764243896536719894486, 7.70120378742242110037791567344, 8.847504543034012959748680300118, 9.371007932926603702345524282267, 10.77745306640211080768639104426, 12.69549999626950664910485252938
