| L(s) = 1 | − 1.43e7·3-s + 1.07e9·4-s − 7.94e12·7-s + 2.05e14·9-s − 1.54e16·12-s + 5.58e16·13-s + 1.15e18·16-s + 2.89e19·19-s + 1.14e20·21-s + 9.31e20·25-s − 2.95e21·27-s − 8.53e21·28-s − 1.27e21·31-s + 2.21e23·36-s + 5.21e23·37-s − 8.01e23·39-s − 4.27e24·43-s − 1.65e25·48-s + 4.05e25·49-s + 5.99e25·52-s − 4.15e26·57-s + 4.14e26·61-s − 1.63e27·63-s + 1.23e27·64-s + 4.89e27·67-s − 1.77e28·73-s − 1.33e28·75-s + ⋯ |
| L(s) = 1 | − 3-s + 4-s − 1.67·7-s + 9-s − 12-s + 1.09·13-s + 16-s + 1.90·19-s + 1.67·21-s + 25-s − 27-s − 1.67·28-s − 0.0541·31-s + 36-s + 1.56·37-s − 1.09·39-s − 1.34·43-s − 48-s + 1.80·49-s + 1.09·52-s − 1.90·57-s + 0.687·61-s − 1.67·63-s + 64-s + 1.98·67-s − 1.99·73-s − 75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(31-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+15) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{31}{2})\) |
\(\approx\) |
\(1.522730885\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.522730885\) |
| \(L(16)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p^{15} T \) |
| good | 2 | \( ( 1 - p^{15} T )( 1 + p^{15} T ) \) |
| 5 | \( ( 1 - p^{15} T )( 1 + p^{15} T ) \) |
| 7 | \( 1 + 7945347009886 T + p^{30} T^{2} \) |
| 11 | \( ( 1 - p^{15} T )( 1 + p^{15} T ) \) |
| 13 | \( 1 - 55850658624240986 T + p^{30} T^{2} \) |
| 17 | \( ( 1 - p^{15} T )( 1 + p^{15} T ) \) |
| 19 | \( 1 - 28940493759797289098 T + p^{30} T^{2} \) |
| 23 | \( ( 1 - p^{15} T )( 1 + p^{15} T ) \) |
| 29 | \( ( 1 - p^{15} T )( 1 + p^{15} T ) \) |
| 31 | \( 1 + \)\(12\!\cdots\!98\)\( T + p^{30} T^{2} \) |
| 37 | \( 1 - \)\(52\!\cdots\!14\)\( T + p^{30} T^{2} \) |
| 41 | \( ( 1 - p^{15} T )( 1 + p^{15} T ) \) |
| 43 | \( 1 + \)\(42\!\cdots\!14\)\( T + p^{30} T^{2} \) |
| 47 | \( ( 1 - p^{15} T )( 1 + p^{15} T ) \) |
| 53 | \( ( 1 - p^{15} T )( 1 + p^{15} T ) \) |
| 59 | \( ( 1 - p^{15} T )( 1 + p^{15} T ) \) |
| 61 | \( 1 - \)\(41\!\cdots\!02\)\( T + p^{30} T^{2} \) |
| 67 | \( 1 - \)\(48\!\cdots\!14\)\( T + p^{30} T^{2} \) |
| 71 | \( ( 1 - p^{15} T )( 1 + p^{15} T ) \) |
| 73 | \( 1 + \)\(17\!\cdots\!14\)\( T + p^{30} T^{2} \) |
| 79 | \( 1 + \)\(50\!\cdots\!02\)\( T + p^{30} T^{2} \) |
| 83 | \( ( 1 - p^{15} T )( 1 + p^{15} T ) \) |
| 89 | \( ( 1 - p^{15} T )( 1 + p^{15} T ) \) |
| 97 | \( 1 + \)\(19\!\cdots\!86\)\( T + p^{30} T^{2} \) |
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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
−18.57275632553021938505355857439, −16.48775551524834734048459406654, −15.81425429637375516354980959483, −12.94940695721070938537734426185, −11.48028575238635141664831400282, −9.947083637710997274185675936201, −6.96740059321038252636747650880, −5.87235848615511777518285612127, −3.25573448259648831663093913685, −0.956208883679535076544701928471,
0.956208883679535076544701928471, 3.25573448259648831663093913685, 5.87235848615511777518285612127, 6.96740059321038252636747650880, 9.947083637710997274185675936201, 11.48028575238635141664831400282, 12.94940695721070938537734426185, 15.81425429637375516354980959483, 16.48775551524834734048459406654, 18.57275632553021938505355857439
