L(s) = 1 | − 1.34e8·2-s + 3.76e12·3-s + 1.80e16·4-s − 5.04e20·6-s − 2.41e24·8-s − 4.40e25·9-s + 5.86e27·11-s + 6.77e28·12-s + 3.24e32·16-s − 3.33e33·17-s + 5.90e33·18-s − 6.70e34·19-s − 7.87e35·22-s − 9.09e36·24-s + 5.55e37·25-s − 3.84e38·27-s − 4.35e40·32-s + 2.20e40·33-s + 4.47e41·34-s − 7.92e41·36-s + 8.99e42·38-s − 2.55e43·41-s + 2.42e44·43-s + 1.05e44·44-s + 1.22e45·48-s + 4.31e45·49-s − 7.45e45·50-s + ⋯ |
L(s) = 1 | − 2-s + 0.493·3-s + 4-s − 0.493·6-s − 8-s − 0.756·9-s + 0.447·11-s + 0.493·12-s + 16-s − 1.99·17-s + 0.756·18-s − 1.99·19-s − 0.447·22-s − 0.493·24-s + 25-s − 0.866·27-s − 32-s + 0.220·33-s + 1.99·34-s − 0.756·36-s + 1.99·38-s − 0.727·41-s + 1.91·43-s + 0.447·44-s + 0.493·48-s + 49-s − 50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(55-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+27) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{55}{2})\) |
\(\approx\) |
\(0.9328173593\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9328173593\) |
\(L(28)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + p^{27} T \) |
good | 3 | \( 1 - 3761108326126 T + p^{54} T^{2} \) |
| 5 | \( ( 1 - p^{27} T )( 1 + p^{27} T ) \) |
| 7 | \( ( 1 - p^{27} T )( 1 + p^{27} T ) \) |
| 11 | \( 1 - \)\(58\!\cdots\!42\)\( T + p^{54} T^{2} \) |
| 13 | \( ( 1 - p^{27} T )( 1 + p^{27} T ) \) |
| 17 | \( 1 + \)\(33\!\cdots\!46\)\( T + p^{54} T^{2} \) |
| 19 | \( 1 + \)\(67\!\cdots\!22\)\( T + p^{54} T^{2} \) |
| 23 | \( ( 1 - p^{27} T )( 1 + p^{27} T ) \) |
| 29 | \( ( 1 - p^{27} T )( 1 + p^{27} T ) \) |
| 31 | \( ( 1 - p^{27} T )( 1 + p^{27} T ) \) |
| 37 | \( ( 1 - p^{27} T )( 1 + p^{27} T ) \) |
| 41 | \( 1 + \)\(25\!\cdots\!38\)\( T + p^{54} T^{2} \) |
| 43 | \( 1 - \)\(24\!\cdots\!86\)\( T + p^{54} T^{2} \) |
| 47 | \( ( 1 - p^{27} T )( 1 + p^{27} T ) \) |
| 53 | \( ( 1 - p^{27} T )( 1 + p^{27} T ) \) |
| 59 | \( 1 - \)\(12\!\cdots\!38\)\( T + p^{54} T^{2} \) |
| 61 | \( ( 1 - p^{27} T )( 1 + p^{27} T ) \) |
| 67 | \( 1 + \)\(16\!\cdots\!46\)\( T + p^{54} T^{2} \) |
| 71 | \( ( 1 - p^{27} T )( 1 + p^{27} T ) \) |
| 73 | \( 1 + \)\(40\!\cdots\!94\)\( T + p^{54} T^{2} \) |
| 79 | \( ( 1 - p^{27} T )( 1 + p^{27} T ) \) |
| 83 | \( 1 + \)\(69\!\cdots\!54\)\( T + p^{54} T^{2} \) |
| 89 | \( 1 + \)\(63\!\cdots\!42\)\( T + p^{54} T^{2} \) |
| 97 | \( 1 - \)\(77\!\cdots\!74\)\( T + p^{54} T^{2} \) |
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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
−11.43657029815464726005165978227, −10.53598224191326206245050661503, −8.892690188553175985043801263409, −8.620263487685319478041583599104, −7.03994885982393591134180786788, −6.06652282971988591681831534566, −4.20401637334274635167179408894, −2.69493769422471681644267809698, −1.96400619117936034411607030301, −0.46238611772833055801104884995,
0.46238611772833055801104884995, 1.96400619117936034411607030301, 2.69493769422471681644267809698, 4.20401637334274635167179408894, 6.06652282971988591681831534566, 7.03994885982393591134180786788, 8.620263487685319478041583599104, 8.892690188553175985043801263409, 10.53598224191326206245050661503, 11.43657029815464726005165978227