| L(s) = 1 | + 16.1·2-s + 243·3-s − 1.78e3·4-s + 3.91e3·6-s − 9.67e3·7-s − 6.18e4·8-s + 5.90e4·9-s − 7.23e5·11-s − 4.34e5·12-s + 1.96e6·13-s − 1.56e5·14-s + 2.66e6·16-s − 1.06e7·17-s + 9.52e5·18-s + 1.37e7·19-s − 2.35e6·21-s − 1.16e7·22-s − 1.61e7·23-s − 1.50e7·24-s + 3.17e7·26-s + 1.43e7·27-s + 1.73e7·28-s + 6.49e7·29-s + 1.96e8·31-s + 1.69e8·32-s − 1.75e8·33-s − 1.71e8·34-s + ⋯ |
| L(s) = 1 | + 0.356·2-s + 0.577·3-s − 0.873·4-s + 0.205·6-s − 0.217·7-s − 0.667·8-s + 0.333·9-s − 1.35·11-s − 0.504·12-s + 1.46·13-s − 0.0775·14-s + 0.635·16-s − 1.81·17-s + 0.118·18-s + 1.27·19-s − 0.125·21-s − 0.482·22-s − 0.523·23-s − 0.385·24-s + 0.523·26-s + 0.192·27-s + 0.190·28-s + 0.588·29-s + 1.22·31-s + 0.893·32-s − 0.781·33-s − 0.648·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(2.096196096\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.096196096\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 243T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 16.1T + 2.04e3T^{2} \) |
| 7 | \( 1 + 9.67e3T + 1.97e9T^{2} \) |
| 11 | \( 1 + 7.23e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 1.96e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 1.06e7T + 3.42e13T^{2} \) |
| 19 | \( 1 - 1.37e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 1.61e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 6.49e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 1.96e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 1.21e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 1.00e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.38e9T + 9.29e17T^{2} \) |
| 47 | \( 1 - 4.48e8T + 2.47e18T^{2} \) |
| 53 | \( 1 + 8.69e8T + 9.26e18T^{2} \) |
| 59 | \( 1 - 7.87e8T + 3.01e19T^{2} \) |
| 61 | \( 1 - 5.77e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 1.15e10T + 1.22e20T^{2} \) |
| 71 | \( 1 - 1.12e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 2.17e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 5.93e9T + 7.47e20T^{2} \) |
| 83 | \( 1 - 4.31e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 1.65e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 3.63e10T + 7.15e21T^{2} \) |
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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
−12.68111490795258051284827658435, −11.12701065877742951850082725348, −9.885708440841169041387198274432, −8.798651913775138577298249806594, −7.959155142311002678319156833393, −6.27497896750777196437163905580, −4.94081983124696263401021124535, −3.78036938884725356035143428143, −2.57867713454472472401733495254, −0.72834284510988799290535024381,
0.72834284510988799290535024381, 2.57867713454472472401733495254, 3.78036938884725356035143428143, 4.94081983124696263401021124535, 6.27497896750777196437163905580, 7.959155142311002678319156833393, 8.798651913775138577298249806594, 9.885708440841169041387198274432, 11.12701065877742951850082725348, 12.68111490795258051284827658435
