| L(s) = 1 | − 8.19e3·2-s + 3.05e6·3-s + 6.71e7·4-s − 2.49e10·6-s − 5.49e11·8-s + 6.76e12·9-s + 3.05e13·11-s + 2.04e14·12-s + 4.50e15·16-s + 1.37e16·17-s − 5.54e16·18-s − 8.11e16·19-s − 2.49e17·22-s − 1.67e18·24-s + 1.49e18·25-s + 1.28e19·27-s − 3.68e19·32-s + 9.30e19·33-s − 1.12e20·34-s + 4.54e20·36-s + 6.64e20·38-s − 1.83e21·41-s + 2.92e21·43-s + 2.04e21·44-s + 1.37e22·48-s + 9.38e21·49-s − 1.22e22·50-s + ⋯ |
| L(s) = 1 | − 2-s + 1.91·3-s + 4-s − 1.91·6-s − 8-s + 2.66·9-s + 0.883·11-s + 1.91·12-s + 16-s + 1.38·17-s − 2.66·18-s − 1.93·19-s − 0.883·22-s − 1.91·24-s + 25-s + 3.18·27-s − 32-s + 1.69·33-s − 1.38·34-s + 2.66·36-s + 1.93·38-s − 1.98·41-s + 1.69·43-s + 0.883·44-s + 1.91·48-s + 49-s − 50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(27-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+13) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{27}{2})\) |
\(\approx\) |
\(2.931788804\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.931788804\) |
| \(L(14)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{13} T \) |
| good | 3 | \( 1 - 3051358 T + p^{26} T^{2} \) |
| 5 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 7 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 11 | \( 1 - 30510390062894 T + p^{26} T^{2} \) |
| 13 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 17 | \( 1 - 13720682212698242 T + p^{26} T^{2} \) |
| 19 | \( 1 + 81175017677663554 T + p^{26} T^{2} \) |
| 23 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 29 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 31 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 37 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 41 | \( 1 + \)\(18\!\cdots\!86\)\( T + p^{26} T^{2} \) |
| 43 | \( 1 - \)\(29\!\cdots\!14\)\( T + p^{26} T^{2} \) |
| 47 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 53 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 59 | \( 1 - \)\(11\!\cdots\!58\)\( T + p^{26} T^{2} \) |
| 61 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 67 | \( 1 + \)\(32\!\cdots\!78\)\( T + p^{26} T^{2} \) |
| 71 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 73 | \( 1 - \)\(33\!\cdots\!38\)\( T + p^{26} T^{2} \) |
| 79 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 83 | \( 1 + \)\(10\!\cdots\!62\)\( T + p^{26} T^{2} \) |
| 89 | \( 1 + \)\(27\!\cdots\!14\)\( T + p^{26} T^{2} \) |
| 97 | \( 1 + \)\(38\!\cdots\!54\)\( T + p^{26} T^{2} \) |
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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
−15.20665397032226332036803527751, −14.34298902994141365380939546284, −12.53811648245079469234051846733, −10.29678843616309040328704113720, −9.070917313127442811700080056427, −8.193979394052157457158331367200, −6.86051414376156613880663155661, −3.74473329041877878222748452745, −2.43942041833936374294974320011, −1.24278843282188874623982059963,
1.24278843282188874623982059963, 2.43942041833936374294974320011, 3.74473329041877878222748452745, 6.86051414376156613880663155661, 8.193979394052157457158331367200, 9.070917313127442811700080056427, 10.29678843616309040328704113720, 12.53811648245079469234051846733, 14.34298902994141365380939546284, 15.20665397032226332036803527751
