L(s) = 1 | + 3.87e8·3-s + 6.87e10·4-s + 2.75e15·7-s + 1.50e17·9-s + 2.66e19·12-s − 1.73e20·13-s + 4.72e21·16-s − 1.13e23·19-s + 1.06e24·21-s + 1.45e25·25-s + 5.81e25·27-s + 1.89e26·28-s + 1.10e27·31-s + 1.03e28·36-s − 3.32e28·37-s − 6.70e28·39-s + 2.82e28·43-s + 1.82e30·48-s + 4.94e30·49-s − 1.18e31·52-s − 4.37e31·57-s − 1.83e32·61-s + 4.13e32·63-s + 3.24e32·64-s + 2.62e32·67-s − 6.03e33·73-s + 5.63e33·75-s + ⋯ |
L(s) = 1 | + 3-s + 4-s + 1.69·7-s + 9-s + 12-s − 1.53·13-s + 16-s − 1.08·19-s + 1.69·21-s + 25-s + 27-s + 1.69·28-s + 1.57·31-s + 36-s − 1.96·37-s − 1.53·39-s + 0.111·43-s + 48-s + 1.86·49-s − 1.53·52-s − 1.08·57-s − 1.34·61-s + 1.69·63-s + 64-s + 0.354·67-s − 1.74·73-s + 75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(37-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+18) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{37}{2})\) |
\(\approx\) |
\(4.037108943\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.037108943\) |
\(L(19)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - p^{18} T \) |
good | 2 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 5 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 7 | \( 1 - 2757049053441698 T + p^{36} T^{2} \) |
| 11 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 13 | \( 1 + \)\(17\!\cdots\!42\)\( T + p^{36} T^{2} \) |
| 17 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 19 | \( 1 + \)\(11\!\cdots\!18\)\( T + p^{36} T^{2} \) |
| 23 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 29 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 31 | \( 1 - \)\(11\!\cdots\!82\)\( T + p^{36} T^{2} \) |
| 37 | \( 1 + \)\(33\!\cdots\!42\)\( T + p^{36} T^{2} \) |
| 41 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 43 | \( 1 - \)\(28\!\cdots\!98\)\( T + p^{36} T^{2} \) |
| 47 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 53 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 59 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 61 | \( 1 + \)\(18\!\cdots\!38\)\( T + p^{36} T^{2} \) |
| 67 | \( 1 - \)\(26\!\cdots\!18\)\( T + p^{36} T^{2} \) |
| 71 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 73 | \( 1 + \)\(60\!\cdots\!62\)\( T + p^{36} T^{2} \) |
| 79 | \( 1 + \)\(89\!\cdots\!78\)\( T + p^{36} T^{2} \) |
| 83 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 89 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 97 | \( 1 + \)\(11\!\cdots\!22\)\( T + p^{36} T^{2} \) |
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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
−17.25396695958024090027761339847, −15.23073282031725027497282147837, −14.36827860214655035688255079722, −12.14878350499604638457258071217, −10.48365263593876863416160988116, −8.353847216045962597627841715128, −7.16263114430842817672262374421, −4.70743504801183611876567229226, −2.60718706792983963651012216456, −1.57524017151587483408727010259,
1.57524017151587483408727010259, 2.60718706792983963651012216456, 4.70743504801183611876567229226, 7.16263114430842817672262374421, 8.353847216045962597627841715128, 10.48365263593876863416160988116, 12.14878350499604638457258071217, 14.36827860214655035688255079722, 15.23073282031725027497282147837, 17.25396695958024090027761339847