| L(s) = 1 | − 512·2-s − 3.26e3·3-s + 2.62e5·4-s + 1.67e6·6-s − 1.34e8·8-s − 3.76e8·9-s − 3.54e8·11-s − 8.56e8·12-s + 6.87e10·16-s + 1.19e11·17-s + 1.92e11·18-s + 3.35e11·19-s + 1.81e11·22-s + 4.38e11·24-s + 3.81e12·25-s + 2.49e12·27-s − 3.51e13·32-s + 1.15e12·33-s − 6.13e13·34-s − 9.87e13·36-s − 1.71e14·38-s + 5.22e14·41-s + 1.00e15·43-s − 9.28e13·44-s − 2.24e14·48-s + 1.62e15·49-s − 1.95e15·50-s + ⋯ |
| L(s) = 1 | − 2-s − 0.165·3-s + 4-s + 0.165·6-s − 8-s − 0.972·9-s − 0.150·11-s − 0.165·12-s + 16-s + 1.01·17-s + 0.972·18-s + 1.03·19-s + 0.150·22-s + 0.165·24-s + 25-s + 0.327·27-s − 32-s + 0.0249·33-s − 1.01·34-s − 0.972·36-s − 1.03·38-s + 1.59·41-s + 1.99·43-s − 0.150·44-s − 0.165·48-s + 49-s − 50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{19}{2})\) |
\(\approx\) |
\(1.005235481\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.005235481\) |
| \(L(10)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{9} T \) |
| good | 3 | \( 1 + 3266 T + p^{18} T^{2} \) |
| 5 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 7 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 11 | \( 1 + 354349618 T + p^{18} T^{2} \) |
| 13 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 17 | \( 1 - 119842447106 T + p^{18} T^{2} \) |
| 19 | \( 1 - 335013705758 T + p^{18} T^{2} \) |
| 23 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 29 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 31 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 37 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 41 | \( 1 - 522162604887122 T + p^{18} T^{2} \) |
| 43 | \( 1 - 1000250360894414 T + p^{18} T^{2} \) |
| 47 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 53 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 59 | \( 1 + 10228968070290322 T + p^{18} T^{2} \) |
| 61 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 67 | \( 1 + 50467064407716994 T + p^{18} T^{2} \) |
| 71 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 73 | \( 1 - 60615173082969074 T + p^{18} T^{2} \) |
| 79 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 83 | \( 1 + 352960460737558306 T + p^{18} T^{2} \) |
| 89 | \( 1 - 487058048464217618 T + p^{18} T^{2} \) |
| 97 | \( 1 - 1501067319528053666 T + p^{18} T^{2} \) |
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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
−17.25015686684452779350115640818, −16.09913007655302117059442623329, −14.42937327358380382208673063191, −12.13732071521618368298974150525, −10.77559796819373836412693834240, −9.164228427777453638680081946230, −7.62096215390452891726412479010, −5.77303505402709931288202008215, −2.87088902024946567099645033173, −0.852338956263912012290963447041,
0.852338956263912012290963447041, 2.87088902024946567099645033173, 5.77303505402709931288202008215, 7.62096215390452891726412479010, 9.164228427777453638680081946230, 10.77559796819373836412693834240, 12.13732071521618368298974150525, 14.42937327358380382208673063191, 16.09913007655302117059442623329, 17.25015686684452779350115640818
