L(s) = 1 | − 4.43e38·3-s + 5.84e48·4-s − 4.63e68·7-s + 1.96e77·9-s − 2.59e87·12-s − 8.62e88·13-s + 3.41e97·16-s − 7.39e103·19-s + 2.05e107·21-s + 1.71e113·25-s − 8.71e115·27-s − 2.70e117·28-s + 1.24e121·31-s + 1.14e126·36-s − 1.52e127·37-s + 3.82e127·39-s − 3.52e132·43-s − 1.51e136·48-s + 1.34e137·49-s − 5.04e137·52-s + 3.27e142·57-s + 4.73e144·61-s − 9.11e145·63-s + 1.99e146·64-s − 1.63e148·67-s − 1.25e151·73-s − 7.58e151·75-s + ⋯ |
L(s) = 1 | − 3-s + 4-s − 1.63·7-s + 9-s − 12-s − 0.0508·13-s + 16-s − 1.94·19-s + 1.63·21-s + 25-s − 27-s − 1.63·28-s + 1.97·31-s + 36-s − 1.44·37-s + 0.0508·39-s − 1.72·43-s − 48-s + 1.66·49-s − 0.0508·52-s + 1.94·57-s + 1.15·61-s − 1.63·63-s + 64-s − 1.99·67-s − 1.47·73-s − 75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(163-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+81) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{163}{2})\) |
\(\approx\) |
\(0.8891374927\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8891374927\) |
\(L(82)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + p^{81} T \) |
good | 2 | \( ( 1 - p^{81} T )( 1 + p^{81} T ) \) |
| 5 | \( ( 1 - p^{81} T )( 1 + p^{81} T ) \) |
| 7 | \( 1 + \)\(46\!\cdots\!14\)\( T + p^{162} T^{2} \) |
| 11 | \( ( 1 - p^{81} T )( 1 + p^{81} T ) \) |
| 13 | \( 1 + \)\(86\!\cdots\!26\)\( T + p^{162} T^{2} \) |
| 17 | \( ( 1 - p^{81} T )( 1 + p^{81} T ) \) |
| 19 | \( 1 + \)\(73\!\cdots\!62\)\( T + p^{162} T^{2} \) |
| 23 | \( ( 1 - p^{81} T )( 1 + p^{81} T ) \) |
| 29 | \( ( 1 - p^{81} T )( 1 + p^{81} T ) \) |
| 31 | \( 1 - \)\(12\!\cdots\!62\)\( T + p^{162} T^{2} \) |
| 37 | \( 1 + \)\(15\!\cdots\!74\)\( T + p^{162} T^{2} \) |
| 41 | \( ( 1 - p^{81} T )( 1 + p^{81} T ) \) |
| 43 | \( 1 + \)\(35\!\cdots\!86\)\( T + p^{162} T^{2} \) |
| 47 | \( ( 1 - p^{81} T )( 1 + p^{81} T ) \) |
| 53 | \( ( 1 - p^{81} T )( 1 + p^{81} T ) \) |
| 59 | \( ( 1 - p^{81} T )( 1 + p^{81} T ) \) |
| 61 | \( 1 - \)\(47\!\cdots\!22\)\( T + p^{162} T^{2} \) |
| 67 | \( 1 + \)\(16\!\cdots\!34\)\( T + p^{162} T^{2} \) |
| 71 | \( ( 1 - p^{81} T )( 1 + p^{81} T ) \) |
| 73 | \( 1 + \)\(12\!\cdots\!46\)\( T + p^{162} T^{2} \) |
| 79 | \( 1 + \)\(60\!\cdots\!42\)\( T + p^{162} T^{2} \) |
| 83 | \( ( 1 - p^{81} T )( 1 + p^{81} T ) \) |
| 89 | \( ( 1 - p^{81} T )( 1 + p^{81} T ) \) |
| 97 | \( 1 - \)\(12\!\cdots\!06\)\( T + p^{162} T^{2} \) |
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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
−10.11783999944372795033525005120, −8.593167337047669282123797531781, −7.04069697896008390529952709666, −6.53756309319070024788768014086, −5.93903918882746854997198435093, −4.62876655275776987041328147602, −3.45523036334612448573078309614, −2.54473923250732832187444871326, −1.44714423338764113914967176894, −0.34909254111996443773051951042,
0.34909254111996443773051951042, 1.44714423338764113914967176894, 2.54473923250732832187444871326, 3.45523036334612448573078309614, 4.62876655275776987041328147602, 5.93903918882746854997198435093, 6.53756309319070024788768014086, 7.04069697896008390529952709666, 8.593167337047669282123797531781, 10.11783999944372795033525005120